should I suspend a floor-standing turntable?

IMO, if the floors are truly bouncy, putting the TT on an end
table will not help. It must be wall mounted, which is really
no big deal. Try this test - take a half glass of water and put
it on the table where you plan to mount the TT. If the water
wobbles significantly when you walk normally, the table isn't
going to work. If the water stays pretty still, you're OK. IMO,
heavy wood or granite does not stop the vibrations caused by
springy floors - assuming they fail the water test. BTW, I
disagree with TBG about wall mounting not helping. Yes, walls
vibrate too, but not like floors. Good luck.

The floors of a high-rise building are often worse than the suspended timber floors of a house for Structure-Borne feedback (and that's the only feedback that you need to worry about)...😫
Your safest bet is as close as possible to a structural column or the core containing the fire stairs and lifts if that is possible.
Being in Southern California.....the building will have been designed for earthquake loading with flexible steel columns and beams and probably prestressed concrete floors....😱
Good for staying alive....but the worst recipe for audio happiness unfortunately....😢
Good luck....🏢

Wood blocks can be found at lumber stores cheap and granite slabs can be found in monument shops -- work out the best price there. You don't need one size of monument for sure :-)

After you select one of the above options, if there is still any uncertainty whether you have overcome all manner of vibrations, there is one more inexpensive step that I have found to complete the job.
Try positioning three or four Cryo'ed hardened steel compression springs named "Cryo Baby Promethean Mini Isolators" (from Machina Dynamica) between the turntable bottom and the support surface of wood or granite. And/or try a set beneath the wood or granite, itself.
You are now better protected, and may proceed to engage in safe sound.
Listening to some favorite recordings will tell you whether each step you take toward better isolation is actually delivering more natural and satisfying audio in your system.