Commanders fill different roles depending on a faction. For all factions they give small and cool abilities that benefit an army, but for some of them, namely SOV, OKW (especially in live version) and USF (to lesser extent) they also fill the gaps in core army or give some units that are powerfull enough to make these gaps managable.
For SOV, such gaps are lack of elite infantry and lack of handheld AT. For OKW, lack of mg and easily accessible AI infantry. For USF its mines.
One could say that this is bad design but in fact it is not the case. These 3 factions are the aggressive ones, so they get commanders that dramatically change the way faction plays and can surprise the opponent, paying with the fact they usually need to pick the doctrine first. On the other hand, OST and UKF, being defencive factions have most things in their core so they don't need to pick that fast, but they pay with predictable unit composition and, in theory, smaller bonusses.
This basically means there is no real way or need to ballance around core. On the other hand this design is not always followed cousing big differences between commander usability.
I feel the most important thing is to ballance faction commanders between each other, so I would change your idea to test factions with the commanders that are used the least in automatch and tournaments, instead of no commanders at all. Giving them some buffs is necessery but nobody really knows what kind of buffs becouse they are either played against meta commanders or not at all. Playing them out against equal opponents could really show how to make them shine. I also strongly belive that with all commanders being more or less equal there will be no point in ballancing core.
Soviets are lacking far more sadly, like an MG nest by default for example, a good medium tank, something which the current T-34/76 is not filling the role of and so forth.
OKW on the other hand is really only lacking an MG by default but that is an integral part of a core Army, no other Army in the history of CoH has only had a doctrinal MG which was not part of the core Army. And I don't really see how Obers, Sturms and Jaegers aren't "easily accesable" AI infantry, not to mention Falls, but then again I do agree that pretty much all of them are expensive (if you consider 300 man power for the Sturm's performance expensive) and 2 of them are doctrinal but still, Obers are Elite AI infantry while Sturms are good at doing whatever as long as you keep them safe and alive which is a bit hard to do since it's a 4 man squad compared to the British RE when upgraded.
UKF is only lacking a mobile in-direct fire weapon by default, which brings up the point you made which is very well put, that doctrines make these gaps manageable but still, paying fuel for something which is supposed to be there already... I just don't see the point, better to make the mortar pit a defensive position/emplacement which can garrison 2 mortars instead of having them spawn with/in it from the get go, just give them some sort of boost/bonus once inside with vet, like the vickers, instead of this immobile emplacement which everybody and their grandmothers are whining about, besides the bofors. And if you're asking if it's possible, yes, it is, I have done it because Svanh told me how, and I knew it was possible before since Europe at War, a very old and big mod for CoH that is centered around realism, has also done this, by using the mortar pit as an MG nest in which the MG can rotate a full 360 degrees to engine targets, like in a normal building.
The USF is lacking practical and well performing units on the other hand, at the current moment, their MG comes too late to be doing the job it should be doing, which is killing and suppressing infantry thus halting a blob/infantry advance because of some better light AT capabilities which are none-existent atm, I mean, the MG42 costs less, comes out wayyy earlier and can do what the .50 cal does even better. While the Pack howitzer is underperforming compared to the leig for example because of the crew required being 3 and the debuff they get which I think is a bug since all USF team weapon crews get it which makes them easier to kill off, that's why the suggestion of the mortar being added to them, that and the alternative to rifle rifle rifle opening.
As for Wehr, all they need is some balance love, and something which both Soviets and Wehrmacht lack, a forward retreat point and perhaps field repairs, Wehr more so than Sovs because the Soviets at least have 3 doctrinal options for repairing in the field which are present in most of their doctrines.
That's mostly it, I think we can pretty much guess most of the problems, and maybe you knew them and I'm just repeating myself or someone else but I just wanted to post this so the OP takes it into consideration of what each Army is "lacking" in it's core, sometimes severely.
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