My basement ceiling is plastered so I can't
see any piping except for what I tore up yesterday.

I'd say that's part of your problem. Having a ceiling in the basement
is pointless.

I recently used a mechanical drain cleaner (The Rigid tool one that
Home Depot sells for ~20-something bucks; I think it's 25 feet long).
I used it 5 times and got clumps of hair 3 times and nothing 2 times.
So you might need to try it a few times or use a longer one as someone
else already suggested.