I will not start using 2FA authentication because I dislike google and Google Authentication. You better think some other way to enable 2 factor authentication if you want to force it to your customers.
Ignoring the rest of the post, but this is a bit inaccurate. The "Google Authenticator" uses a very common open-source algorithm to determine the 2-factor-codes. The app from Google is simply the most commonly used, which is why the whole system is often named after Google. There are however alternative implementations for all kinds of platforms and the algorithm is simple enough that any semi-decent programmer can create his own personal implementation, so that you don't have to trust any third party software producer.