Azure DevOps Artifact.zip folder structure for .NET Core 3.0 application
I'm looking this fact:


  I have an ASP.NET Core 3.0 application for which I have a full CI/CD setup in Azure DevOps. The Application is hosted on a Ubuntu 18.04 server machine.


Therefore I can safely assume that you are developing ASP.NET Core 3.0 app to be hosted in Ubuntu. Any .NET Core 3.0 (or later) application means that you should rely on the dotnet build instead of using VSBuild. 

Also you had stated that you will host the app on Ubuntu 18.x, then you should also run the build on Azure DevOps agent that runs on Ubuntu. This means you should only use dotnet build in DotNetCoreCLI@2 task, because VSBuild task only runs on Windows based agent, not Ubuntu and it is intended to compile .NET Framework and other platform other than .NET Core.

Please consult the official doc of DotNetCoreCLI@2 task at https://docs.microsoft.com/en-us/azure/devops/pipelines/tasks/build/dotnet-core-cli?view=azure-devops

After you use dotnet build, use dotnet publish to publish your artifact. It is much easier and it's also the best way to publish .NET Core apps.

I have sample usage of dotnet build and dotnet publish of this on my repo: 
https://github.com/eriawan/dotnetcore-cicd-samples/blob/master/yaml-samples/sample_pipelines_check.yml