I've checked with our developers on the issue once again - unfortunately, they have confirmed that this is expected behavior that occurs due to the fact that only one RDP session can be running on Windows 10 machines at any given time and in this case, it cannot be used along with a console session (and Remote Utilities works in a console session). For example, please see this Microsoft Q&A page. I apologize if this makes our program less useful for you.
So you are actually saying that it is impossible to "Shadow" a Windows 10 machine with your software, although it works sometimes, and all the time with other software? Your software can not be used to support a user who is RDP'ing into a Windows 10 machine? I still believe it is a misunderstanding. I am NOT trying to log on as a second user - I am only interested in seeing and interacting with the screen of the already RDP-logged on user. The Microsoft Q&A page you linked to confirms that it is possible in Windows 10, and allowed from a licensing point of view as well, but you say that your software does not support this.
I understand your situation and I'm very sorry for the inconvenience, but I simply forwarded the information I've received from our developers on this issue. In fact, I've tried reproducing your scenario as well, but it only works exactly as our developers stated - after I've connected to a remote machine via RDP, I also connected to the same machine via Full Control or via the View Only connection modes and when I switch to the Terminal session in my Full Control window, it simply kicks the other user out of the RDP session. I've tested this with both Windows 10 and Windows 7 just in case.