That might really limit your options as Quicksync is pretty much the gold standard for efficiency and performance. A discrete GPU works too but you might have trouble finding them in a Chinese mini PC.

I used AMD for years (RX580) but it wasn’t always the best quality and having a separate GPU really sapped a lot more power. I don’t believe ARM processors are really optimized for transcoding either. Under the hood, Jellyfin, Emby, and Plex all use ffmpeg to do their transcoding so maybe you can research more about its performance with each platform.

Again, used is always an option. You’re not giving money to any of these companies by buying used parts and it sounds like this would get you more ‘horsepower’ from your budget than buying a bunch of new stuff.