Dear Pipedown, thanks so much for your reply, your idea is very
interesting. Could you please explain a little bit more about how to
built my own new valve seat using silicone RTV and how to make sure to
have a very even edge? Thanks so much!!

PipeDown wrote:

Hopefully the floor is tiled up to the pipe and the toilet sits on top of
the tile. In which case just clean the floor and caulk to the new shape.
With a toilet that old, the floor may have been done any number of ways.
Try flapper valves with different shapes and material. Silicone or rubber
might work better than the other. You might be able to build your own new
valve seat using silicone RTV but you will have to have a very even edge.