Those plastic wedges may reduce squeaking and rattling while driving, but I wouldn't call them "structural". They're just small pieces of plastic and would surely get crushed if they were intended to stop body flex while jacking.
When working on the front, I drive onto ramps instead of jacking up the car whenever possible. When using a jack, I try to lift one end of the car evenly on both sides, though it's not possible to do in the front unless I had two floor jacks. (I jack-up the pumpkin in the rear sometimes, but I don't think I'd jack up the front by the crossmember.) When lowering the front subframe onto jackstands, it is certainly unnerving to see the entire car flex.
My dad used to tell me stories of his '67 Sport Fury convertible. When he would jack up one corner, the rear fender would go "sproing" and collapse! When he let it down, the fender would pop out, back into shape as if nothing ever happened. I imagine that would be a big problem if there was ever any bodywork done in the rear. Jacking it up would've probably caused the paint to crack and any bondo to pop out.