I'd prefer that you fix your plaster as needed. 1/4" drywall to me is just a bandaid and before doing that, I would start one room at a time and tear off the plaster and lath and put on drywall on the studwalls. You can reinsulate and upgrade any electrical at the same time.
The wall you can push tells me there is some broken lath or rotted stud behind it so that part may need replacing anyway.
Are you going to do this yourself? Repairing plaster isn't easy but doable but then I can't see how good or bad your walls are. Good luck. It also always helps to have a helper/significant other who has some handyman skills.