When you exit out of a non-Console RDP session, the OS stops rendering the desktop. Same as minimizing a window. There's nothing for OBS to capture, as nothing is being processed by the OS.
Using the console login switch MIGHT let this work, if Windows feels like it, and if you just disconnect the session instead of logging out. No guarantees though, as this really isn't what OBS is meant to be used for. VNC may possibly be a better option over RDP (assuming along with the requisite encryption for security), if it keeps the desktop active.
Necro'ing a thread from last year that was previously necro'd from two years ago isn't a great idea though, in any case.