The .net core project indeed runs fine on Linux in AWS. It's a great success story for us.
The ASP .NET projects I maintain have a budget that doesn't easily allow us to just migrate them to .net core. There are some hairy legacy libs and code in there that would also prove challenging to resolve too.
There's also the question of SQL server. It's not so much of an issue when you're dealing with Azure, but SQL server on Linux isn't ready for prime time yet IMO.