I have 6 windows rigs, and manage all from one computer. You can remote desktop without a monitor plugged in. Claymore updates take me less than 5 minutes for all 6. With remote monitor, life is easy.
Everything is possible. Makes only sense what's convenient and what's the price.
On Linux I expose only ssh secure port, and use text terminal from my phone via slow GPRS connection (for instance) to get work done.
On Windows I have to run GUI via fast link. I don't like the idea to expose RDP ports to the Internet, so I either have to use a kind of VPN, ssh passthrough, or use tools like TeamViewer. RDP has issues - miner sometimes can't read/manage temps/fans if started from remote session. Sometimes it works - I can't find out why and when. For miner there is absolutely no sense to run GUI to manage rigs.
It is a matter of taste, but Linux is also free and more stable, has a watchdog that can reboot it even if GPU driver hangs. Windows sometimes can't or require external watchdog device. Etc, etc.
I don't remote from outside of my network, but when I did, I just set up one external remote to main, and then vnc from there, not as fast but worked. Would be easier with teamviewer. To see temps, you need to get an HDMI dummy plug ($15). If hang, it can be harder to fix automatically, but I just run stable... If Claymore gets gpu error, best to have it run a restart.bat file instead of trying to reset miner, it always hangs system. So there are some drawbacks, but it is kind of more preference for me. I don't want to spend extra time setting up drivers, making more scripts and such. I may switch in future, but it is just easier for me for now.