Does MarkStein Publisher run on macOS Catalina and Big Sur?

Yes, MarkStein Publisher from version 7 is running on macOS 10.13 to 10.15 and 11.x.

MarkStein Publisher from version 6.5 is running on macOS 10.12 to 10.15. Older versions up to 6.4 are running on macOS 10.11 to 10.14.