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.

Is the browser somewhat up to date? In case Chrome doesn't work, try Firefox or Kiwi.

But it's quite possible that 4.x is so far outdated, some necessary system routines are simply missing and can't be replaced.

Even tubular (alternative youtube app) is only backwards compatible to minimum 5.0 -> https://github.com/polymorphicshade/Tubular

Make sure that all system dependencies such as Google play services are running the latest compatible version -> https://m.apkpure.com/google-play-services/com.google.android.gms (most likely you'll have to go to "old versions -> show all" and then try one version per year until you can narrow it down to something that works).

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.