Same. For most people in our space, a Dell Optiplex micro is more than enough horsepower (except few expansion ports). There’s no reason to be running a legacy server consuming 2.5kW just to host some movies or store your camera footage.
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.
For Frigate you can just add a Coral TPU to offload the heavy lifting but with Jellyfin you’ll likely need a GPU/iGPU (Intel) that can handle transcoding. I don’t know what specific models you’re looking at but you might try to determine their passmark score. I think it’s recommended at least 2000 score for a single 1080p transcode. Alternatively you can stick to specific file formats that won’t need transcoding but this might be some work.
You could get a used Dell Optiplex micro. They have i3/i5 processors and can be found for around $100. This will at least get you going and you can always build something later.
It’s “powerful” in the sense that it has 48 PCI lanes and can use almost 800GB of memory (ECC included). This is just way overkill on most homelabs so the extra power draw isn’t worth it.
Are you doing this all manually or using the *arr suite? For me, this process takes a minute or two depending on the size of the files with Proxmox and ZFS but even previously on Windows 10 with SnapRAID it was quick.