Docker as mentioned above is great especially if you're going down the homelab rabbit hole. You'll find it really useful to get familiar with for future projects.
Another alternative is to install DietPi. It's an operating system for a raspberry pi. It has all these programs you mentioned with install scripts including docker and installing (and updating) is as easy as checking a box. Also has a pretty good community in case you run into an edge case.
eBay is your friend, Shield came out about 9 years ago so lots on the used market. Also the phoning home for the paranoid can also be handled. Since it's been around for so long there's not a lot left someone hasn't already figured out. Maybe not the perfect answer but worth considering. As a media player it just works and it's tried and true.
Have you looked into maybe getting a shield and changing out the launcher? It's a great emulator and will play any video file thrown at it. Worth looking into if you can't find any sbc alternative.
Ah falling down the rabbit hole. If you're only looking to transcode videos stay away from the graphics card. Look into Intel quick sync. Newer Intel cpus can transcode over a dozen 4k streams at a time using little power. However the next issue depends on what operating system you want to use, Windows, Linux, docker container or proxmox to run Plex/emby/jellyfin. They all come with their own issues/caveats to actually getting quick sync to work. However each is well documented. Learning is the fun part. I went down a similar route ended up with a nuc12pro running proxmox running a whole bunch of stuff but primarily used for Plex running in an lxc container. They are small powerful and don't use a lot of power and importantly also well documented.
Yep alot slower. You can Google it but you aren't the first person to have this problem.
I've seen this issue come up often. Do not use the ethernet connections on the TV as they are frequently 100mbit even on brand new TVs and mostly likely the source of your bottleneck. Either use the wifi or as some suggested an Nvidia shield and hook that up to Ethernet.
When I was in Atlanta and I initially saw her campaign signs I honestly thought someone vandalized them with cartoon glasses. It's fairly distracting especially since she is supposed to be very competent. I hope she does well in her election but she needs to get new photos.