I think you are misunderstanding the YouTube reference. It's likely saying to use a YouTube video to validate you've setup the app properly. You don't use it for authentication.
There are a handful of authenticator apps like Google Authenticator, LastPass Authenticator, Authy, Okta Verify and so on. Google Authenticator is definitely the weakest of the bunch.
I'd actually recommend that you use LastPass or Authy for one particular reason. As part of the authenticator app setup, once things are done, you'll be given an option to download/save about 6 permanent codes. You should grab them and store in a secure location as these codes bypass the 2FA.
Now if your phone were to ever break or app stops working, these codes is how you get back in.
However, if you do transition to a new phone, the problem with Google Authenticator is that it doesn't save anything anywhere other than your phone, so you have to go and re-do the setup on every site.
LastPass and Authy, on the other hand, have a user account that backs up all of your codes so you can move from device to device and get them all back without losing access.
You should still always grab the permanent codes from sites and save them, and as mentioned do not put them in a publicly accessible location as that defeats the 2FA entirely.
Now, as far as notifications go, a handful of apps support 2FA through their own app, like the GMail app. But this is very specific to a site that has it's own app and so on. Since this isn't that situation, it doesn't apply. And YouTube doesn't have it's own authentication, it comes under your Google account umbrella.