Synchronization can be start of nightmare.
2-factor authentication application must be an independent one and don't need to be synchronized with your Google account. If Google account is hacked, you will lose many information there includes 2FA backup.
Use Aegis, an open source 2FA. Google Authy is closed source.
https://getaegis.app/You can also turn off the 2FA synchronization in Google Authenticator if exporting your accounts is too much of a PITA.