Go to the gaming machine and upgrade it to > home or edu version if needed. Enable remote desktop with auth on your network for that machine. As long as Windows is booted and able to be logged in to on the gaming machine you can go on your other machine:

Win+R mstsc.exe and put in the gaming machine's name or IP and follow the instructions, checking all "remember this" boxes (there's 2, three if you count the certificate).

RDP won't let you play games but it is functionally identical to sitting at the machine itself.

I meant to come back and fix this but missed the window. I am unsure if the home/edu/N/P whatever versions of windows 7, 8, 10, and 11 support actually remote controlling the desktop as opposed to merely getting a "video feed" as it were. There's ways to upgrade to pro that are beyond my pay grade to discuss, but i think you can get a clean copy of windows that supports RDP for $130. If your alternative is "having a monitor plugged in 24/7" or "dummy cables but still have to plug a monitor in if something goes wrong with steam link {and it will. -ed.}" or other hacks/hardware, and you're already running windows at least the GPU/gaming side, RDP practically pays for itself even though it's $130 for that feature.

Someone else probably has alternatives (moonlight? bazzite? gopro and a soviet-era robotic arm (it only leaks a bit when it's hot outside.))

consider a different remote access option i.e. sunshine + moonlight instead of using steam remote play

Can confirm, Sunshine + Moonlight are a killer combination.

I run a Windows VM on one of my servers for some gaming because I don't run Window's otherwise, and with Sunshine on the VM, I can play with moonlight from my TV, laptop, desktop, phone, ROG Ally (Bazzite), tablet, basically anything that can support Moonlight.