MVC and Web Forms are ASP.NET applications. The same reason you do not want to continue with Web Forms applies to ASP.NET MVC. The .NET 4.X framework is not going away anytime soon. To put this into perspective, I support classic asp web apps built in the late 90s running on the latest Windows OS. Maybe upgrading is not necessary?
If you decide to upgrade, for any reason, then .NET Core is the clear choice.