I have a ASP.NET Web API controller that provides a saveAndNew operation.
This class is derived from System.Web.Http.ApiController as expected.
Application I
You might want to try removing/commenting out TransferRequestHandler in ApplicationInsights.config file.
GitHub issue "REST API app can have all requests filtered out" here states that: "commenting out the one line in the ApplicationInsights.config caused successful requests to start getting logged. Web API 2 project.
<!--<Add>System.Web.Handlers.TransferRequestHandler</Add>-->