damaged stair tread - repair options?

I would glue and screw as you seem to be suggesting, try to clean out any debris in crack to get a nice tight fit. Then use wood putty on top if needed, sand lightly, stain, etc. I think this will be easier than trying to match the appearance with a new piece. If you do decide to replace, bring the split piece to lumber yard. Trouble is, lumber available today may not match that from 1920.

I had a creaky questionable staircase in my '50s house when I moved in. I did as you suggest and approached it from the bottom. I used 2 X 4s and joist hangers (the metal things that are used to hang blind joists) and screwed it all together with a cordless drill. You can gain a little upward pressure as you screw the hangers in by slanting the screw up a little. Mine are carpeted and I have no idea what is under it, but they are solid and squeak free now.