We have a few .net core (2&3) web apps using aurelia. Some are using the 2 method approach where we run 2 terminals; dotnet run or dotnet watch run for CS changes and au run for aurelia. Using a bat file is fine too and we have those as well. We also have some of the apps using in .NET app.UseWebpackDevMiddleware() in startup.cs. It runs your .NET app and clientside webpack and will rebuild and refresh on clientside changes all with a dotnet run or by running debug with Visual Studio.
There are a few github slns you can grab for starting points, use the aurelia CLI to build a new dotnet/aurelia project, and here’s a post talking about using HtmlWebpackPugin in .net core 3.
Having it built in to your c# side makes it easier for non frontend devs to kick off a local dev build by using the terminal or clicking the debug button in Visual Studio. My own preference is having it separate as I have more control over what’s running and how it’s running.