Questions.
I'm an irrational Anglophile and have often presumed that if the 13 Colonies stayed in the British Empire, British North American colonies would still grow tremendously in extent, population, and agricultural/industrial potential, leadng to a British Empire that for all intents in purposes included the OTL USA in it, giving Britian access to huge american resources when necessary (such as in a major war, etc)
How likely is this? Isn't equally equally possible that a defeated/stillborn American independence movement would lead to a situation in which Britain deliberately keeps its north american colonies under-populated and less intensively developed so as to not complete with Britain? Could this lead to a situation in which a British Empire incorporating virtually all of north america would find itself unable to draw on the support it could have had from an independent USA in a hypothetical ATL "WW1" against the NeoNapoleonic Confederation or the Austro-Ottoman Hegemony?
So, forget the feel-good democratizing factor an independent USA may have been, maybe an independent USA would have been good for Britian in terms of geopolitics as well.
That's a...very complicated question. There's a lot of different ways it could've gone so deciding the most likely path that history would've taken isn't that easy.
First off, to address the standard conclusion that I think just about everyone on this board accepts, if the Colonies ever turned their mind to rebellion, then they are effectively gone. Britain might win the first revolutionary war, but if the colonists decide that they aren't happy with the way the British are handling things then it's kind of in their nature that sources of discontent will keep bubbling up until a second rebellion starts, and a third, and so on. One of those days, Britain is either going to lose the war, or realise that the cost of militarily pacifying the Colonies is exorbitant and the very act of pacifying the Colonies is causing more anger amongst the population. In the end, one way or the other, the British would retreat from the situation. There are times when you just have to accept that you can't win.
So we're talking here about a situation where things never get that bad, which means a certain amount of removing troublesome factors and a certain amount of appeasement. We're going to have to go with some sort of Articles of Confederation plan here, it just won't work otherwise. To me, the most likely result would actually see the Colonies divide into three separate chunks - the New England Colonies in one Confederation, the "Deep South" (and maybe NC?) in another and those in the middle forming the third. Not only does this allow the three Congresses to deal more with "local issues" - i.e. it saves the New Englanders from having to wrack their consciences over what the southerners are doing with their slaves - it also helps London break down the powerful American "bloc", thus meaning that if they do something a bit stupid, or, say, finally decide it's time to end slavery, then they can actively enlist the support of much of the Colonist population without turning it into a war against the entire continent. For the record, I'm not really sure how slavery would be "solved" in TTL, except I think it would be messy but fixable. That's one of the reasons that your question is so loaded.
As for the issue of keeping the Colonies depopulated - not really. It's just not viable. The British were hardly capable of turning away migrants, and even if they were to try to, the governments of the Colonies were in favour of all the influx of population so you'd be faced with a situation where the British would have to virtually seize control of a port militarily to enforce such a policy. That ain't gonna happen, so the demographic increase will continue. As for nerfing the US economy, you kind of have a point, though. As Bob said above, the British in this period were heavily protectionist towards their nascent industrial economy back in the mother country, and so they actively forbade industrialisation of the Colonies as it threatened to put businesses back home into bankruptcy, especially as the Colonies were home to many of the resources that the industrialists needed, and thus they could potentially outproduce the British businesses for less money. The British economy of this era was built on the idea that resources flooded into Britain and goods flooded out. Anything to upset this balance would've been disastrous. Now, eventually this must change, but much later than OTL. Maybe you'd be looking at some industrialisation allowed in the 1860s - perhaps even the Cotton Gin would give the whole movement some impetus - with a more widespread liberalisation by the 1890s or so. In consequence you're looking at the Americas being economically weaker, but the British Isles being stronger. The corollary to all this is that all those migrants have to go
somewhere, so while you only have American industry developing very late, you have a much, much, much bigger growth in other sectors such as farming and resource collection. You're also looking at a society that has grown to very large levels (population-wise) while being dependent on the powerhouse production of a land across the sea. This means that you may well see the American economy create conditions of its own that we have never seen OTL - for instance, any goods that come into the Americas must arrive on the East Coast ports, so maybe you see all those extra workers developing a very impressive series of redistribution, where middlemen race each other to deliver the latest clothing, furniture and tools out to the in-land markets. If Britain was a nation of shopkeepers, maybe America would be a nation of couriers, middlemen and auctioneers? I don't know for sure, I'm just ad-libbing, but it's possible.
Regarding not being able to draw upon the Colonies for support during a war: no. That's pretty unlikely. If the Colonies aren't willing to contribute to a war, or the British aren't willing to recruit them, then that means the society is already in a state where the Colonies divorcing themselves of Britain is imminent. We can probably assume that the American suspicion of standing armies lingers for far longer, since they have no need to create an army of their own in TTL, and any issues with the system can be projected onto the British, thereby continuing to fuel distrust of paid armies. However, once a world war comes along - and it won't start in 1914, but one will happen eventually - then we must assume that the Americans will contribute. What's more, they will most probably
ask to contribute. This will be a world where the Americans retain some of the British sensibilities - such as seeing anyone who dominates Europe as being "too powerful" and needing to be knocked down a peg or two - and so it's inevitable that any American who feels even a little patriotism and reads the paper will read of Britain entering a world war, and when it becomes obvious that this is one that is draining the mother country of troops, there will be many men who want to go over and expend their naive pluck wasting their lives in the trenches. You'll see militia units being converted into combat battalions for deployment in Europe, for sure. This is a world where Britain will indeed command more soldiers than OTL, and will have a stronger economy, so we can only assume that balance of power dictates its enemies will be somewhat stronger, so we can't just assume that this means that Britain can automatically win a world war with its zombie-like victory through overwhelming numbers, but they will be large and they could well make a difference. On the other hand, with the British now controlling a far larger American state, the chances of there being a new front on the American border with either Spain's colonies, or their independent successors, is high, so that will change things. The interesting questions that come out of this, again, is "will the frontier spirit and Manifest Destiny spur the Colonists to demand expansion southwards if the war is won?" and "will the experience of the war make standing armies more popular or less?" Those are both very interesting questions which could have huge knock-on impacts on the future of society, but they are the kind of questions which there are just too many answers to. Any slight change in the way the war goes could change the answer to those questions too. It's too hard to read into while staying with the general case.
As for the "feel-good democratizing factor", this will of course be a world where Britain dictates the terms of the end of the war, not America, so you're looking at a situation where the regimes aren't going to change. Britain always supported democracy, and it will take the opportunity to step in and prop-up any breakaway democracies that form from the remnants of its enemies' empires, but it also had a fierce support for legitimate governments, so there's not going to really be any policy developing of attacking Johnny Foreigner to get rid of his corrupt and evil dictatorship. America can still pride itself on being the Land of the Free, but that concept is going to revolve around the idea of "we are free and you wish you were like us" rather than "we are free and you are free because we made you free".
As for the existence of the US being good for British geopolitics - the existence of the US has mainly been beneficial to the British in that it created a proxy for them. When it didn't want to expend its own efforts on cleaning up areas it considered beyond salvation, the Americans always seemed a little too willing to jump in and get themselves dirty, even though (especially in the early years) it often meant failure and disgrace. When things went wrong they could wash their hands of a situation. When things went right, they may not entirely have agreed with the specifics of what the Americans decided on, but at least they could always agree that it was better than what existed before. America has, in other words, been good for British geopolitics in the way that having a robotic housemaid is good for a millionaire. It may be a little too advanced for its own good, its programming may break down every once in a while and it probably comes with a risk of overthrowing its masters in a glorious robot revolution a few decades down the line, but it's just so much easier than employing ten actual humans to do the same job only slower, and unlike with humans you don't even have to get your hands wet keeping them happy and answering their questions and resolving their issues. It just gets on with it by itself.
Other issues I haven't even mentioned here, but it's almost 3am and I've been typing for most of an hour now: you're still going to need to find ways to sort out the Indian tribes, who the British were by-and-large more supportive of than the Americans, and that means holding the leash of the Colonists when they get restless and want to expand a bit further west. You're also going to have to deal with the pressure some time in the 20th century for the Colonists to have more parity and not come under London's sway. You'd need to decide whether the ARW never happening meant that Colonies such as Australia, Alt-Canada if it still exists independently of the Colonies (Ruperts Bay Colony, maybe?), South Africa etc are going to want to go independent eventually or if the lack of a convincing independence revolution means that the Colonists will simply never be content with rule from London because they don't know/are scared of the alternatives. You're going to have to decide what London does about the filibustering attempts in Central America - does it slap them down for disturbing the natural order of things, or is it pressured into allowing the Colonies to keep expanding their influence, and perhaps even political power?
So essentially what I'm trying to say here is yes, it could be done, but there are so many factors to consider. If you can envision a sliding scale of loyalty/productiveness/engagement in the empire, ranging from "wants to go their own way" to "more British than the British themselves" you could probably find a way of hitting literally every point on that scale...but to do so, you need to find a way past all the little pitfalls. It's possible, but it takes thought, and everyone's thoughts are different so it's really hard to be more accurate than this.