I tried many screen recorder software due to some work requirements, free or paid, over a dozen. Unfortunately, there's no such UI in my memory. So i'm sure it's not a top-ranked or say mainstream app. But this feature you've mentioned seems to be rare. I know OBS is able to do that (maybe a little bit complex than the one-clicking operation you've said), Snagit allows you to switch between screen and webcam, RecMaster and many tools are capable of recording PiP effects and give you freedom to hide or show webcam during the capture.....
Anyway, let us know if you've found this app out. I still want to have a try