This is the official method provided by Apple to download legacy apps. It only works if you have previously "purchased" (downloaded for free counts as purchased) a version of YouTube while using your current Apple ID on a newer device in the past.

to download. If a compatible version exists in Apple's database, a pop-up will ask if you want to "Download an older version of this app".

For a few hours, yes. But then you will encounter:

Note: Even if this installs, the app may fail to load content due to outdated API connections. 3. Third-Party "Legacy" App Stores (Advanced) Some users in the "retro tech" community use tools like Checkmate, Store!