You can also turn off the 2FA synchronization in Google Authenticator if exporting your accounts is too much of a PITA.
I did not understand what you meant here. You mean it is possible to turn off the synchronization on Google authenticator if you do not want to export the authenticator codes that you setup already? You do not need the online synchronization before you can export your Google authenticator account. As for recommendation, any 2fa application that backups online should not be used.