I believe it is a micro 3070. Amazon link just for example, but very easy to find under $250 and very low power consumption. https://www.amazon.com/Dell-OptiPlex-3070-Desktop-Computer/dp/B07TTCJB2P?th=1

I’ve had similar requirements for a few years. Finally settled on 8 bay synology with a dell micro optiplex. I have a 10gb card in the synology and actually use a usb to 2.5gbE in the dell with zero stability issues. Some synology products can transcode a stream or two of 4k with quick sync but not many of the 8 bay ones. I have a 9th gen i5 with 32gb of ram that handles all my transcoding needs. Done 2x 4k transcodes with no issues while also streaming other 1080p for additional users.

Nothing to add except calling the PSP retro made me feel old.