if most of service currently just have Google 2FA on their services?. So far, I'm just seeing it on service i use no other 2FA options except SMS/email verification.
I'm not entirely sure what you mean here, perhaps because I do not use any Google products so I'm not aware of what their 2FA options are. If a site such as a crypto exchange says "Scan this code with your Google Authenticator App", then you should be able to use any 2FA app. I have certainly done this in the past and it works fine.
If the service only offers SMS or email verification for 2FA, then you obviously can't use an app - you can only use what the site offers. Both of these are not great choices, but you can make it slightly better by using a different email address with a different password to the one you use to log in to the account, or by using a burner phone with a number you do not use for anything else and which you never use to access the services in question.
The issue with that is if you store the password to the service in question in the same password manager. If someone can access both your password and your 2FA by compromising a single source - in this case, your password manager - then your 2FA isn't really a second factor at all, it is both factors rolled in to one.
I had to check and correct timer on my devices to help 2FA works again. I don't know reasons why timer was broken like that. Could you explain it, please.
I can't explain the specifics, but the way that most 2FA apps work is to take the current time, round it down to the nearest 30 seconds, and use that along with the shared secret to generate a code. Therefore, the code will change every 30 seconds as the time updates every 30 seconds. If the clock on your device and the clock on the service are out of sync with each other by 30 seconds, then the code you generate will always be different to the code the service is generating, and so they will never match until you resync your timer.