first when you go to method hotspots or response time hotspots on service level, do you see those methods there? If not it is possible that this part of code is custom and Dynatrace has problem with detecting it. Did you try custom service detection? If you will define executions of those async tasks as another service it should be visible in single purepath.
I just tried this and for me an, async POST http request to an API Controller works fine with latest OneAgent version.
If your problem still exists, I'd recommend opening a support ticket, so we can take a closer look at that.