The only two Google apps I can't seem to get away from are Authenticator and Maps. I travel a lot, and I find the reviews are helpful for deciding where to go. Even if I don't read the reviews, the number of reviews are helpful to find a useful store of any time.
Apple maps' reviews aren't as helpful.
Of course this doesn't matter it all if you aren't going to use backup, in which case I can't think of any reason not to use Authy. As far as in-app TOTP backup goes you may not need it. I'll get to that later.
A good TOTP app for Apple devices if you want something that can backup to iCloud is OTP Auth. It can save/restore/sync with iCloud. OTP Auth also supports making and restoring from encrypted backup files which you can save using the usual share options.
Both Authy and OTP Auth support Apple Watch which is handy. You might not actually need in-app TOTP backup. When I set up a TOTP on a new site I save a copy of the QR code PNG on my Mac. I keep the collection of such QR codes in an encrypted file (which then automatically ends up in my Time Machine backup and my offsite backup). If I need to set up a new TOTP authenticator or restore an old one I simply decrypt that file, open all the PNGs files, and scan then.
If you don't have a large number of accounts that is not much trouble. I think I have maybe 20. I open the PNGs all in Preview, set it to show one per page, then it is scan, page down, repeat. Takes a second or two per code. It would probably get a bit annoying around 100.
Even if your TOTP app does support backup and restore and you use that, I think it is still a good idea to save the codes as described above, because if you ever want to change TOTP apps you likely can't restore from your previous app's backup. If you've saved the individual codes there is no such limitation.
That's what I do. I consider the encrypted file of PNGs to be my main backup, and the encrypted backup I saved in iCloud Files from OTP Auth to be for restoring OTP Auth if I ever have to reset a device or setup OTP Auth on a new device.
If you still have Google Authenticator somewhere, such as on the device you upgraded to your new phone from, you may be able to get your TOTP data out of it and into whatever TOTP app you use on the new phone. See my other comment that is near this one for details on how to do that.
That mechanism gives you a QR code containing the account information for one or more accounts, which you scan on the new device in Authenticator to import those accounts. Repeat until all accounts are migrated. I'm not sure how many accounts they can put in one QR code (I've only used Authenticator briefly to help someone else figure something out, and only put in 4 codes), but I'd expect at least 20.
To make a local backup just use the migration mechanism but instead of scanning the QR codes on a new device save them somewhere. On Apple devices you should be able to simply screenshot the migration QR codes. I've heard that on Android Authenticator blocks screenshots.
The contents of the QR codes are base64 encoded Google protocol buffers. If you extract the contents of the QR code, you can use the command line tool otpauth [1] to extract the TOTP setup URLs for each of the sites in the Google migration code.
You can have otpauth turn those individual site setup URLs into QR code images. This will do the trick:
otpauth -qr --link <text_from_google_QR_code>