The rotating 6-digit code that is generated by the app on your phone does not run out. The number is based on a formula that uses a secret number (specific to the one account and known only to your app and the service that validates the code) and the time. So, as long as time keeps ticking, new codes will be generated.
I use old phones as 2FA code ...
It does not defeat 2FA. If someone steals only your password, or if you use the same password everywhere and it leaks, the second factor will still protect your account.
That's the purpose of 2FA: add another layer of security. Having both on the same device does not change that. That "one-password-for-everything" is still secured by 2FA anyway.
As far as I can see from the respective websites those apps are specific to one service.
I am not aware of a generic "push" protocol. So if you want to get that form of push notification from a service you will have to install the matching app.