A lathe will open the opportunity to more easily experiment with shapes and sizes. However, it takes some practice to get matching halves and it can be expensive by the time you buy all the tools, as well as the lathe. It’s very fun though.
A hole saw for a drill will produce matching halves with a hole in the center for the axle. You can still bevel the edges and mess with shapes but not to the extent and ease of a lathe. Much cheaper than a lathe.
My grandmother and great grandmother made yoyos for me out of a tree branch. They would just find a branch of a nice diameter, drill a hole in the end, close to center, cut a few cross sections, put a dowel in the hole, and I would have the easiest, most wobbly, vibey yo on the planet. I wish I still had a few.