If you remove the ability for the truck to push, also remove it from All scout cars and light vehicles. There is no reason a Heavy Halftrack shouldn't be able to push infantry when a scout car can, it already takes damage from small arms, and a single shock squad can kill it in about 3 seconds while it's setting up.
a) The properties of an OKW building under setup have literally zero bearing on a halftrack's ability to push infantry.
b) Every scout car and light vehicle in the game must be purchased with fuel and manpower, and take time to accumulate fuel and build, literally none are available in the first crucial engagements when fuel is capped and map control gets carved out. I cannot wrap my mind around the fact you are trying to tie the elimination of this abusive strat to nerfing all other factions' light vehicles.
c) "Pushing with it is risky, exposes it to small arms fire" - so does pushing with light vehicles, which require teching, cost fuel and manpower, are not immediately available on the field, and whose counters can hit the field by the time they are on the field.
sWS IS COMPLETELY FREE AND IMMEDIATELY AVAILABLE. Until you realize how this makes sWS pushing qualitatively different to all other light vehicles in the game, there is no point in discussing it any further.