Android 4.1 is super old, so getting anything still updated to run on it could be complicated. And because of that, I have maybe two or so ideas, but I can't ensure their viability. First, to replace the phone's Android system for a newer one. Second, to try to use a custom player, like Revanced or Grayjay. And as an extension of the second idea, if no programs you can find work, to try to manually compile them against the latest API version that supports Android 4.1.

https://f-droid.org/packages/org.schabi.newpipelegacy/ NewPipe Legacy works on Android v4.1.X, although of course it won't allow them to log into their YouTube account, as it does not use the YouTube API.