Why not just backup your password to your email and maybe a notepad app?
2FA sometimes fails.
What if it doesn't sync right or the 2FA app gets a bug like it happened with WinAuth, wouldn't that be a worse fate?
What if you lose that number?
I like the current setup. It's easier and even newbies can easily understand how things work.
A lot of people aren't so good with technology. So less complications are better.

The idea of linking accounts... Don't think it's advisable, as you never know when someone might change.