I'm currently a frontend engineer. I told my team lead a while back that I'd like to transition to the server side, especially since our server team is woefully understaffed and our frontend team is overstaffed. He told me "Sure but you'll need to learn all that stuff on your own time". So I never did it because I'm interested in doing other things at home.
It's always couched in some doublespeak about "ownership". Trust me, if I were owner or part-owner in the enterprise, I wouldn't mind. But I'm not.
People complain about entitlement attitudes among the young. They need to look at employers if they really want to see entitlement.
It sounds like your boss feels entitled to your time without taking an ownership stake in your professional development.