One thing to note on this is that after a week, Google told me that my account might be compromised, severed the link between Gmail and my Proton account, and forced me to log back in to my Google account on my computer.
I reconnected to Proton and it did it again a couple weeks later. Not sure if it's fixed. (or working as intended)
Gmail Forwarding is the old feature that just forwards mail, and that seems to work fine without ever kicking you off, it's only Gmail Connection (which allows sending emails from your Proton mail and having them send from your Gmail address) that's done it for me so far.