They might not port Mono's WinForms to .NET Core? Given how strictly tied to Windows WinForms is, they may not want to give enterprises they are encouraging to use .NET Core any confusion on the subject. Taking that extra step of trying to run the application on Mono shows at least some awareness of possible compatibility issues in the software.

That said, this may still be a great opportunity for .NET Core to encourage trying Mono in its documentation for enterprises looking for cross-platform support. (Hopefully they already knew about Mono by this point, but an extra push won't hurt.

I think Mono Winforms is a total rewrite of the Windows Winforms. There is no shared code.

Sure it is but why not use that code to let .Net Core support WinForms on Linux the way it is going to support it on Windows?

I think you have that a little backwards. This is not up to the Core developers but to the Mono developers. There is nothing preventing the Mono developers from supporting .Net Core.

Aren't these the .Net Core developers who sort of import (or re-implement?) WinForms support from the classic .Net Framework on Windows?

It's shared with WINE I believe. (The native parts)

Wine uses its own Mono instance and I usually run WinForms apps directly with the system Mono set-up without using Wine (although I have both installed).