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.

[sweemer]

Hi @jimwang118 I converted the original discussion into an issue because it seems like there is clearer guidance on the migration now than there was two years ago. If you think that more discussion is required then can you please help to loop in the right people?

[dpogue]

This is especially relevant now in light of the GitHub Actions Ubuntu 24.04 runners dropping Mono and NuGet as part of the standard image, Mono being deprecated, and the NuGet folks encouraging everyone to switch to dotnet nuget