Mechanized infantry is not a defensive weapon.
Yes they are. The doctrine at the time was that the infantry dismounts when fighting begins except for perhaps gunner(s), meaning for the most part they fight just as normal infantry = defensively. The APCs main job is to protect them while on the road away from the front from air, artillery, shrapnel and ambushes.
If you think the mechanized was meant to drive right up to an enemy position while it was still fully loaded with troops all of them firing guns blazing then you have seen too many Hollywood movies or played too many action games. The APC/Mechanized in itself is a defensive tool since none of the carried soldiers can effectively fire their weapons or fight at all when seated in most models, especially not when the vehicle was moving.
Many modern IFVs are also basically light tanks that carry their own infantry support into battle with them
What IFVs are today or became 70 years after the war ended really doesn't hold any meaningful relevance.
As I said, really messy to handle and discourages any switch of divisional equipment type.
No. You said you always end up with deficit of thousands, which isn't the case at all... You never need to go into any deficit at all when swapping in new equipment in a division template, and most of the time I never do so if any of the divisions are actively fighting anywhere.
And it's not messy at all, just require a bit of patience. Simply start production and note in the template how much total stockpile you need from the tooltip. Once stockpile has reached that surplus, then check again and if OK just commit the template change. Like magic, new equipment in divisions with zero shortage.