So it seems to me that the 'HE' issue is 2 fold
One a bigger gun - 2 pounder is never going to cut it
75mm / 3" seems to be the bench mark we should be aiming at - anything lighter is always going to be ineffective
The problem is that tanks in the late 30s could have a large short barrelled gun that was good at lobbing HE/Smoke or a smaller gun that was good at punching holes in things but not a gun that did both.
I would like to see an A10 type tank with twice the thickness of Armor - a more powerful engine to compensate, a 3 man turret with a 13 pounder 9CWT 3" gun capable of firing HE and APBC rounds.
The HE round to use a half charge and for the round itself to 'intrude' into the case to allow for a slower MV, allowing for a thinner walled shell with greater HE content
Same for Smoke.
Oh and for the love of all that is right in the universe - nice big hatches.
In other words a true universal gun tank in 1939!
Now the real problem with British tanks in 1939-41 was not the designs so much, or lack of HE throwing ability...oh no.
It was numbers of tanks, trained men and educated officers who could make use of the weapon platforms and processes and indeed parent formations to command and direct them
In the early years of the desert war the 'come as you are' desert force often fought in hodge podge Brigade sized formations (Jock Columns etc) trying to mix divisional assets into a Brigade slice in an attempt to try and find the right balance of Infantry, Artillery and Armor
What they should have been doing is fighting as a division and even corps formation and making use of all of the concentrated assets at the point of combat
However there was not enough trained men, not enough educated officers with the experience and training to fight at the divisional level in a cohesive fashion.
This was overcome by the addition of UK trained Divisions arriving in theatre during 1942 who had been training as Divisions, and the older units through both experience and in theatre training and finally by Monty insisting that they fought in such a fashion.
So again we come back to having enough people which means expanding the army before late 38 and introducing some form of conscription well before early 39
The lack of trained men and educated officers cannot be overcome by simply improving equipment
Lack of HE shells in a 2 pounder armed tank squadron being engaged by and AT gun screen could easily be rectified by having an attached FO from the Divisions Artillery regiments in an attached armoured FO vehicle calling in fire from Divisional and Corps level artillery to supress
It was found by the end of the Desert war and beyond that a company or Battalion could get support fire from Division and Corps level artillery far quicker than they could often get from their own 3" mortars.
But this level of all arms Cohesion took time, training, practice and often actual war time experience to get right and required more trained men and educated officers (both at Field and Staff level) than the British had to hand in the first couple of years of war.