Again not quite that whole truth..
As you mention
Infantry divisions had their own "cavalry regt", a RAC RECCE regt, until mid war when Recce Corps was formed. The sub units had mainly carriers and a single cruiser sdn.
(Eg 9th Aust inf div had 2/9 cavalry regt). It's role is reconnaissance, including its cruisers.
NZ is an outlier, 4th bde was transformed into an ARMD bde. Like many dominion independent formations, it " hovers" BTW british corps (operationally) and its native division (administrative). But while the history equipped with "cruisers" (shermans), was crewed by many "tank" trained crewman from 1st tank bde back in NZ, it is used as a tank unit in Italy, as their is little scope for true "armoured" warfare.
I was just being mostly compete including the tank armed cavalry Bn. Ofc while the role is recon, they are still tanks, hence the use in Op Bulimba which could count as a cavalry raid. My mainpoint though is the Attatchment of amd Bde to the ID being a clear part of British Doctrine.
The irony of armoured warfare, it is written by converts from outside, and not cavalrymen. (Hobart was RE, Liddell Hart & Fuller Light Infantry, and Guderian Signals.)
RTR was the only units with "real" tanks ( mediums) and laid down many concepts, (eg shoulder aimed guns for mobile aiming in moving vehicles) but like mobile division, never lasted long enough to evolve a true Armoured penetrative, deep battle doctrine. But when cavalry converted, it wanted to play cruiser / horse, and wanted the RTR to play boring slow tank.
Hobart was an old man, but a brain ahead of the times. It was the old horse cavalrymen that were away with fairies. Cavalry charges with horses in modern warfare was dumb, with tanks dumber!
It is the 1941-2 reform of British Army, rewriting training and selection, that drags BA thinking into a modern all arms army. Especially the cavalrymen!!
This is of course nonesense. Unless you are using schoolboys to develop concepts of armoured warfare every officer will have been originally commissioned in another arm of service and ofc there were no Armor officers in the US army during WW2. They were commissioned into another branch.
The reason for the shoulder mounted guns and aiming from moving vehicles is it works. Pre war testing had a 33% first round hit between 500 and 200 yards, and in NW Europe almost all engagements of WW2 were at 500m or less. Until the later 1930s, when the Army in the UK but not Hobo moves away from fire on the move there are no purpose designed AT guns. This is important the field artillery available has a traverse of about 4-6 degrees. So if you can move out of the very narrow frontal arc the enemy has to pick up the trail and move the gun. PAK 36 has 60 degrees, 2lb famously 360.
And being able to fire on the move does not make it mandatory. For the germans halt and fire was mandatory. But the German sights are different. You estimate range lay on the sight and then match the crosshairs on the gun traverse ( which is by spinning wheels) with the sight and bang. Its very accurate particularly at longer ranges where range estimation becomes important. If you are moving the movement of your own vehicle will throw off the sight so the gunner has to be stopped.
The British gun is balanced and light enough to be physically aimed by the gunners upper body, and up to 600 yards the scope is essentially a battlesight lay on and bang after that its graduated. That changes later but then the British don't normally fire on the move later on at AFVs.
The British in fact did develop an clear role for mobile forces thats what the mobile division is for thats why Montgomery refers to a Corps de Chasse and Alamein thats why Compass, why the Pursuit after Alamein goes on for about 1500 miles about double the distance from Warsaw - Moscow, thats why the pursuit after Normandy only stops when it runs out of fuel and at no point do they lose 5-6 whole armies because the enemy reserves can cut them off, the enemy reserves have been destroyed, See also Operation Diadem of for that matter the reconquest of the Ukraine in 43 and Bagration.
But in the end the whole of the British Army is mobile and armoured ( unlike the German or Soviet) and when you refer to Armoured warfare what you are referring to is either bewegungkreig which goes back to maybe 1670 and is most recently codified a couple of years before the Nazis come to power and does not include panzer divisions.
The Trotskite - Zinovievite fascist conspiracy of Deep Battle is not in fact approved in the USSR in 41 and really only used poetically during the war but practiced in the field- and derives from the cavalry experience of the civil war nothing to do with armour per se,
But the concept is basically - Deep Battle requires the first echelon – mainly infantry – directly supported by tanks and artillery to contact the enemy frontage, fixing them in place and preventing reaction to the second echelon – mostly tanks – attacking on a narrow frontage, creating a breakthrough. The exploitation/pursuit force then passes through the breach assaulting the rear echelon.
The failure on the soviet part up to lateish 43 is their failure to prevent reaction to the second echelon resulting in numerous tank armies being destroyed by the German reserves. Its not until they sequence multiple attacks which do suck in german reserves that they are able to launch a deep operation which runs to the end of its ligistics then stops.
To quote the fascist conspirator Tukhachevskii “ In mounting a penetration operation,
the transition from breaking-in battle, to turning movement, must be carefully thought out and adequately planned. These offensive phases must follow one another without any gap in time, let-up in intensity, or hiatus in communication and re-supply.”
One of the differences between the British or Soviet perspective and the German is the Germans want to avoid the breaking in battle by maneuver and surprise that's kinda the whole point.