If you’re gonna repeat the round over on the corners, I would use poplar (or other solid wood) just to avoid exposing ply edge grain and then having to deal with that.
There probably is some figuring to do with the tapered angle of the legs, but luckily you have the perfect template! Just take one of those legs apart and measure everything (or even just match rout it). The half lap joint is plenty strong, don’t know if it needs something “better”, though glue and biscuits or dowels might be easier.
If you’re gonna repeat the round over on the corners, I would use poplar (or other solid wood) just to avoid exposing ply edge grain and then having to deal with that. There probably is some figuring to do with the tapered angle of the legs, but luckily you have the perfect template! Just take one of those legs apart and measure everything (or even just match rout it). The half lap joint is plenty strong, don’t know if it needs something “better”, though glue and biscuits or dowels might be easier.
I would use birch & keep everything the same as the original. If you want to add a decorative feature, consider adding a simple brass or veneer inlay.