I have the same issue with spongy floors. My house was built in the 60s as well. The floor joists they used were too small for the span of the room, so the floors bounce like crazy. So much so, I've moved a huge chunk of my record collection to the basement because I'm paranoid about the floor joists warping. Anyways, I really hate the situation.
I use a Rega RP40, and given the lightweight plinthe and the aluminum brace under the tonearm, it rarely skips, even with 2 toddlers moshing in a circle pit with my horse dog and chihuahua. Granted I have it on a 2x4 Kallax with a bunch of records in it, so the weight likely absorbs some of the vibration. I've contemplated wall mounting as well, but the whole house shakes when someone closes the door. Cheap build on this house.
When I buy a different house, I'll be inspecting the floor build, that's for sure.