Single piece, simple ball turner.

[Toolmaker51]

You have lost me there, I see 3 axis involved either way.

I'm thinking along the set-up, manipulating knee and quill Z's [prior to locking] wherein lathe that's more 'built-in' addressing centerline. Maybe better size control too? Can think of last occasion to ball turn, no-one had the lathe version, used a boring head though. Difficult to get proper finish, heads aren't too great for decent corner radii on cutters. I wouldn't spend effort on form tool even with means to generate accurate cutter, on something like a push rod.