After a few back and force with support, it turns out that the .Net Core APM profiler is not compatible with ARM based linux containers.
It means it is not possible to monitor a .NET application that is running on the official image mcr.microsoft.com/dotnet/core/aspnet:3.1-buster-slim-arm64v8
In AWS, ARM based docker containers are required to be able to deploy a fleet of containers to EC2 instances of type A1 (which are significantly cheaper than x86 machines).