First question is: multiple logins by the domain administrator account? or different users with domain admin privileges?
If giving everyone a shortcut with /admin is not an option, then....
If the former......
Not sure if this will have an affect on your specific scenario, but it has worked for us.
Note: This is old school. But, indeed, a tool that should have never been removed! (In my opinion)
The caveat is that this is no longer included with 2012 and newer. You have to copy some files and two registry keys from an existing 2008 server.
We use this tool all the time, so copying the files and making the reg entries is something we do to every new server that may eventually allow RDP access.
We keep a zipped file in our ScreenConnect Toolbox just for this.
Here's a shot of what we change:

And here is a link with instructions on how to make the old manager work on new servers:
http://woshub.com/how-to-run-tsadmin-msc-and-tsconfig-msc-snap-ins-on-windows-server-2012-rds/