.NET Core was developed with CLI usage in mind and common operations (like migrations/boilerplate generation) can be driven from CLI or Rider, you don't need VS for that.
You may have some problems with 3rd-party libraries. Not all of them have been ported to .NET Standard. I'd recommend checking your project's dependencies in advance. (We did have some problems with legacy COM-based cryptographic libraries and had to build a separate "microservice" hosted under Windows to offload this stuff to. COM stuff works fine under .NET Core, but only under Windows (obviously).)
I develop on Windows with Visual Studio and deploy to “Linux”. We use CodeBuild to compile and create the zip file (lambda) or Docker Container (Fargate).
CodeBuild basically is “Serverless builds”. It launches s prebuilt or custom Docker Container and you just run .net core “publish” command line and it gets all of your Nuget packages and then you run your standard packaging commands.
But Rider should be good. I’m a huge fan of R# and you can cross target Linux, Mac and Windows from either host using msbuild.