It does work, but when you connect, the host computer will actually reconnect its display/relog/reconnect devices when you log in and out, messing with your vmix.

The one last thing I advise against is using the built in windows remote desktop function. If its not make sure to check if your mix one is using the Master bus as audio source. Any video played within vmix will go to whatever bus you assign it to, so as long as your video has its sound turned on (or auto) when its playing on program, it should be in your livestream/website. So in order to monitor the sound you'll need to set up Vmix so that it sends headphones or master to desktop rather then a specific interface. Basically remote desktop programs can send you on the remote side the desktop sound. We recently swapped to anydesk which seems to be lighter and also a bit more stable in terms of resources used. I've found that Teamviewer does use up a significant amount of computer resources.

With most other one-user plans, you can only access one computer. However, no-one will stop you from using it other then a popup that you're trialing or home-use. A big plus for individual users SplashTop’s Solo plan gives you access to 2 computers. Officially speaking you'd need to get a business license to legally use teamviewer for this purpose. The Teamviewer free license is for home use only, ie using your home computer while on the road etc.

Teamviewer vs splashtop forums