Never mind. Two hours after I posted the above, I got an unsolicited text saying "Your Messenger verification code is...." Without doing anything with it, I checked the RCS status in Google Messages and it was turned on (yay!). I also checked my Google Account and the pending auto-verification message was gone. So, it looks like GOS phones WILL auto-verify. And, it looks like that auto-verification is what RCS is dependent on in Messages.
EDIT: I can't find a way to mark this thread solved. Sorry.