That is an i5 1.4ghz 2014 Mini, not an i7, although the the standard model only had a 500gb internal hard drive. Personally I wouldn't pay anywhere near $300 for one of these, they are severely crippled computers. I have one, and although it worked fine as an iTunes server, it was pretty useless for anything else. For example, it would take 40 seconds for System Preferences to open after clicking. This has been discussed extensively around here in the past. Adding a SSD will help, but you can't get around the 4gb RAM issue.
Now, it might be OK for your needs, but I sure wouldn't pay $300, maybe $100? Mine just sits in the closed now, don't have a need for it and wouldn't be worth the effort to sell.
You can use any Mac as a Time Machine destination, you configure this in the sharing control panel although it is so un-intuitive that I had to Google to figure out how to do it.
Regarding video playlists, they work fine on the host Mac (in iTunes on Mojave) and work fine on my two Apple TV's. However, they don't work on in the TV app on my 2018 Mini with Catalina. The playlist is show, but it's blank when you select it. And the TV app on iOS doesn't seem to support video playlists at all, they are not even shown.
Music playlists work fine across all devices however.