Stabilizers

[Imp]

that the whole fire on the move thingie is pretty much overblown

It depends what you are talking about a WW2 stabilizer yes pretty much allowed you to aim in roughly the right direction so could land smoke or HE near by to harass. Stop or slow to fire.

Modern systems fully intedgrated to battle computers depends on the shot ammo you want. Fire from stationary is a must for some esp if delayed fuse ammo so for instance you want to fire through the window or wall & have it go off inside the building.

Firing at armour though so long as its an AP type shell not a HEAT one so high velocity modern vehicles can indeed drive at reasonable speed & turn yet still hit targets at under a mile.
Watch some vids on You Tube some can be quite intresting & show crew quaility as in miss then come on target then off it. Others nail it 9 out of 10 times while whizing around.
There is a great one of a Swedish 9040IFV or whatever its is out of control spining on ice but the gunner still nailed the target traversing through about 120 degrees

Cant remember but I think reason for lack of in M48 was high gun mounting therefore breach block waving around dangerously in the tank.
Sherman was a similar problem I think if not well trained it will have you.
Also dont think it liked humid enviroments a lot.