YoutubeDL-Material is exactly what you're looking for. It does both, either download to the machine accessing the software in the browser, or download to a location on the server. Best of both worlds. It's actually a great bit of software, does more than you expect.
"Just download an YT downloader application" is not what OP was asking for.
I too have been looking for a centralized YT-DLP frontend that I can use to either 1. Store vids on my server 2. Give me (or more importantly, my wife) a prompt to download them.