Amazing. How silly those Frenchies were to surrender when they could easily have continued and beaten the pants off the Germans....
But seriously, the French surrendered because they were beaten. Sure, they had been beaten before they could mobilize their full economic and morale strength but lets not forget the French army was almost as large as the German army, they had more and better tanks and as many aircraft. And they got the stuffing kicked out of them.
In terms of the will to fight, they certainly were beaten (and the discussion is more of a hypothetical about the consequences than an assertion that so-and-so could have led to France carrying on). Petain and the gang thought that Britain would have our necks wrung like the proverbial chicken and it was best to settle for as lenient a peace as possible while it was going. That doesn't mean that France didn't still have various useful resources - its empire and fleet - that could have helped the war effort.
Poland had no empire and no useful fleet; but for the Poles there was no lenient peace going, only national extermination, so they fought on. If they were physically capable of doing it, the French absolutely were.
There wouldn't have been that much to continue a fight. Yes, the French fleet, a couple of squadrons that fled to North Africa and mostly the North African depot troops left there as the best troops had already been sent to France. The French would certainly have tried to ship as many troops off to North Africa as possible but I rather wonder how that would have turned out. For starters, what would the French civilians have done? Meekly waited while the troops were bugging out? If you consider the fleeing civilians OT during the German advance, desperately trying to stay ahead of the Germans, I think you can expect "last helicopter out of Saigon" like scenes.
No troops need escape France at all for the Allies to enjoy the benefits of retaining Indochina, say, or of not having to invade Syria or Madagascar. The French fleet, of course, is a bigger deal than you make it sound like.
And if we can all see the danger to Libiya, the Italians could probably see it too. They almost would have to send their fleet against the troop carriers, whatever the cost.
The cost of sailing right smack into the middle of the western Med, given the availability of both the French navy and the RN, would be the Italians taking a sojourn with Davy Jones. Fine use of resources, that.
The existence of Egypt didn't compel the Italians to make a death-ride against the British fleet. Why should the existence of Tunisia demand a death-ride against an even stronger fleet?
Still, lets assume the French (or at least a French government) makes it to North Africa, ready to continue the war. With what? It would take them years to build up a strong army capable of tackling the Germans. And to equip it properly. The British couldn't help them, they couldn't even equip their own army and needed American material. Even as late as 1943, when the ex-Vichy forces had to be re-equipped, the British couldn't spare the necessary equipment while the Americans could only provide enough for a couple of divisions. In 1943, when the American arsenal of victory was at full steam!
You seem to be assuming that it's now up to the French government in Algiers to save the world single-handed. The USSR is still there. Britain and America have everything they had IOTL. They also have a handy fleet, no need to invade Vichy colonies, and the ability to threaten Libya from two sides - even a small force is useful in this scenario. Japan, meanwhile, doesn't have the staging-point from which it attacked the Southern Resource Area. Clearly, the war is going to be easier for the Allies.
And talking of the American arsenal, lend-lease was largely the result of the shock of the French defeat. If there is no defeat, merely a lost battle of France, would the British and French still get their "free ride". Or would they have had to pay cash, as they had to do before the lost battle? Could they even afford to? Britain practically went bankrupt in 1941.
You answer your own question. If Lend-Lease is still necessary to prevent German victory, then there is absolutely no reason not to grant it. State's pursue strategic goals, they don't start doing things under exact sets of circusmtances like computer programmes; America still wants Nazism defeated expediently and still sees that the Allies need its industrial capacity to do that. And losing France is of course hardly an insignificant setback. In fact, it's a massive defeat that basically excludes the Allies from Europe.
And that is even without considering the political repercussions. Just how eager would the French colonies be to sustain a failed French government (failed in the sense that it is the first duty of a government to defend the nation).
The only colony that revolted against either French government IOTL was Vietnam (against Vichy and for the Allies, although that might well be differant ITTL), so why assume that any other colonies would do?
The British commonwealth largely existed of semi-independent countries with native governments (albeit imported 'natives') excepting colonies like India A decision to support England would have had legitimacy. And even there, the British had to make a lot of concessions which ultimately cost them their empire.
So excepting the vast majority of the empire's population? I find it interesting that British colonies were so much more "legitimate" than French ones: there wasn't yet a Quite Algeria movement.
And those concessions weren't to the dominions but to India and Burma, so the two strands of your argument fail to cross.
The French had mostly old-fashioned colonies and it seems unlikely their native populations would have footed the butcher's bill without due compensation. Which would have meant the end of the French empire. Vichy managed to maintain the French colonies and a part of metropolitan France, as was its aim. How much legitimacy would a die-hard regime have if it merely led to the dissolution of France? Would they even want to try, given the price?
As I said, all the thread asks us to do is suppose that France fights on. Unlike many of the things we are routinely asked to take as read ("the USSR collapses, the Axis win"), this is perfectly possible in a physical sense. Oh, and the Free French had no trouble holding on to their share of the French Empire and were able to draw some manpower from places like Central Africa. Not much, but as I said, strategic positions and the fleet are the main benefits, not manpower and arms.