Not going to happen overnight and even then technically they cannot access your emails because only you hold the password to private keys (if you trust they encrypt your emails with your public key before storing). I prefer keeping my emails local so pop does the job.
Google can access your emails but something like protonmail can't (if you trust them to encrypt your emails).
Sounds good... in theory. Truth is we don't know this, their mail client is proprietary and even if it was open source we still wouldn't know what their servers are logging. I am in a similar situation and I trust Google more than ProtonMail with all that Tesonet data weirdness pointing back to one guy. These shell companies can go and disappear overnight, something to consider.
Trust is multiple things. I "trust" that Google will manage my private Gmail account consistent with my expectations—which are effectiveness and long-term durability. I have no expectation of algorithmic privacy and thus there's no trust to break there.
Emails have a lot of information about my life.
I know what Google is doing with my emails, and as far as I can see Google is not going to die or be purchased by another entity.
With companies like Protonmail and others I just don't know. Even if I trust their current T&C, what happens if they are bought.
Also, while it is equally likely for Google to have rogue employees, I believe they will have more stringent safeguards then smaller companies.
Again, as I said, probably not a very good answer, but more gut driven. And, I say this as someone who is very conscious of privacy.