Could I use this hypothetical desktop app from my desktop, my other desktop, my mobile phone, my wife's mobile phone, and the Chromecast on my media center?
Desktop applications never worked on phones, mostly because Google and Apple want you using their own version of what used to be called "programs". Of course, if you need to download YouTube videos on every device you own and the toaster, then sure, maybe don't use yt-dlp. But, ya know, I'd wager that most people don't need to do that.
Sure. But I think using a web page is a much nicer solution than getting a remote desktop stuffed onto my phone screen and trying to use a desktop-native app that way.