So for a media server, and for power management, I’d ditch the discrete GPU and buy an Intel Core processor 10th gen or newer. Intel Quicksync does well with transcoding media and doesn’t cost you any additional power draw.
Mobo: buy whatever is cheap and compatible with the processor you choose. The more SATA ports the better.
RAM: 16GB should be plenty
Case: depending on what type of form factor you want, you might check out the Node 302 or the Define R7 cases from Fractal Design. These are both designed to hold a ton of HDDs so you can build up a media collection over time and just add a drive or two as needed.
If you don’t plan on growing a collection and prefer to just delete stuff after watching, then you can ignore this and buy whatever case you like.
SSD: 256GB to 500GB is good but more doesn’t hurt.
I just reread your OP and noticed you don’t currently have an SSD. You may consider swapping one in as the boot drive and running the rest of your current hardware (and possibly buying a new case for that “new PC” feeling) since they’re so cheap. This will, hands down, give you the biggest performance boost out of anything.
Realistically, the main benefit you’d get from buying all new hardware is better power efficiency as your current hardware is more than adequate for a media server, but it would possibly take years to ‘break even’ on operating costs depending on how much electricity costs where you live.