No access panel to the plumbing on the back side of that wall? My house is from the same era and has a drum trap. Easy to clean out if you have access to it, but impossible if you try to snake through the tub. That is where I would start.
ETA, I couldn't get my drain ring to move either. You can start with one of those bendy plastic sticks with spikes on them to clear the hair though. Does that drain take an immediate 90 degree turn?