Use dotnet nuget instead of nuget.exe and mono for binary caching

[sweemer]

Migrating this from #22684

I saw the following today at www.mono-project.com:

we recommend that active Mono users and maintainers of Mono-based app frameworks migrate to .NET which includes work from this fork.

This suggests to me that it is finally time to switch the binary caching implementation to use dotnet nuget instead of nuget.exe and mono.

[jimwang118]

Duplicate of #22684.