I use a KVM over IP.
I made my own tiny pilot device for less than £100 and it has worked flawlessly for a few years.
This assumes your server has a hdmi output, if not you will need a VGA to hdmi or similar converter, danger of it becoming a bit of a hodge-podge of adapters, but hey, if it works, it works!
With docker-conpose it took less than 20 mins to setup the system across several machines and have it working. Definitely worth investigating.
Pro tip 1, use it to strip out any languages you don't need and other optimisation techniques like reordering streams will save space and reduce transcoding overheads.
Pro tip 2, You'll encoded fastest if you can offload to a GPU, I did this in two gaming pcs with decent albeit 7 year old, Nvidia cards, but you'll get slightly better results (barely noticeable quality wise, but slightly smaller) through a CPU encode, it'll just takes about 7 to 20 times as long.
If you take the plunge to re-encode, check out the self hosted app tdarr, very handy for large libraries, or watching folders for new files, plus you can distribute the load to other computers if you have any spare headroom.
Encoded over 20tb in a few days with three machines and saved around 6tb of space.
I've no experience with either of those devices. So long as they're comparable in terms of USBC features (otg) to emulate the keyboard and mouse then you should be good as the overhead of running the software seems to be minimal from my usage.
Without that, you'll get video without keyboard and mouse support.