Dedly: This is still an issue, I've reported it, and nothing has been done about it, to my knowledge.
BUT I FOUND A WORKAROUND.
Easy workaround: Use Chrome Remote Desktop or some other RDP client instead.
Harder workaround, but works well...
On the machine you're remoting into where you want to start the Galaxy Client but can't, do the following:
1. Create a new text file somewhere. Doesn't matter where.
2. Open the text file.
3. Add the following:
C:\Windows\System32\tscon.exe 1 /dest:console
C:\Windows\System32\tscon.exe 2 /dest:console
timeout 3
"[path to your]GalaxyClient.exe"
4. Save changes to the text file and close it.
5. Rename the file extension from .txt to .bat (if file extensions are not visible, you'll have to make them visible first in Windows/File Explorer options).
6. Now run the .bat file you just created.
The first two lines will close your RDP session but leave the desktop open (tested in Win7, haven't tried it in Win10 yet). It might be session 1 or 2. It's varied for me in the past, but it's always been one of those two. Put both in there and you're sure to make it work.
The third line tells the script to wait 3 seconds. Might not need this, but I wanted to make sure the RDP session had completely closed before it tried to open the client, just to avoid it failing yet again.
The last line will look something like "C:\Program Files\GoG\GalaxyClient.exe" or wherever you installed the client.
I just tested this, it disconnected my session, and started the client successfully. I RDP'd back into the host machine a couple minutes later and the Galaxy Client was open and waiting for me on my desktop.
Thanks for this workaround, it's not 100% fullproof as the client can still hang when you RDP back in.
Sometimes I have to kill all Galaxy processes and retry, this is on Windows 10.