How would you prove to their data protection officer that you are the owner of that Gmail account?

In theory: they can ask for ID, sworn affidavit, or whatever other means their local laws determine to be valid. At the end of the day, proving that someone owns something is not a new problem. I've also seen "here's some evidence that I know what the contents of the account are, my legal name matches the account and my legal address matches some emails there".

In practice: in my case, anecdotally, they just did it. For some reason owning the backup email account was not enough for the automated workflow to unlock my account, but sending a letter threatening to sue under the GDPR somehow changed their minds.