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Another shout-out for Jellyfin. Somewhat unusual in that it's build with .NET technologies and runs on Linux, but it's proven to be an excellent media server for many years now.

It looks like PeerTube can automatically import videos from YouTube channels.

https://docs.joinpeertube.org/use/channel-sync

And as for letting you or your kids upload additional content, yup that’s what PeerTube is mainly for.

I run a PeerTube instance of my own to host some videos I made. I haven’t tried importing from YouTube to it though. I use yt-dlp to download videos I want to keep.

Jellyfin + yt-dlp/torrents

This is more or less what we do but recently the Jellyfin iPad client has become very flaky. My suspicion is the JS app's memory usage has grown to a point where it's causing the web view embedded in the app to run out of memory in which case getting newer client hardware should fix it.

YouTube archivist is what I would recommend for YouTube videos

Kodi and yt-dlp maybe?