>> Is there a non-core ASP.NET that also lives inside that framework? > Yep!

No, the "non-Core" ASP.Net framework is not ported to the .Net Core platform (i.e. .net 5+). Only ASP.Net Core runs on .Net Core.

This is unfortunate, because it means older web sites based on the classic ASP.Net does not have clear migration path to .net 5+.