How can I affix a ball bearing to a metal shaft without welding?
As much as I like the brazing answer, it's unfortunate that it won't work for your purposes.

A search using the terms "drilled steel ball" resulted in many returns for of threaded and non-threaded balls of various materials. Most of the returns are for larger than 1/4" as you've requested.

If you can adjust your design, there is a source of 8 mm drilled (unthreaded) balls on Etsy. It should be possible to use an epoxy to secure this type of ball to a rod.



Etsy also has a listing for a drilled threaded version.