"This may sound silly, but I consider any application that requests full read access to my Gmail account to be akin to asking for my social security number."
Have you ever emailed your social security number, had your social security number emailed to you, or signed up for any sort of service to your gmail account including your paycheck management company, your tax returns, your bank, or anything else that via email password reset could be used to access your social security number?
Certainly I'd go so far as to say that almost anybody's "real" gmail account could certainly be leveraged to get the last four digits of your social security number. Given the low entropy of the other 5 numbers given other details [1], even if you can say with a straight face that your email has never had your very, very regex-able social security number in it, it's still got most of the bits, most likely. Perhaps not enough to automatically target you without a bit more machine learning than I think we quite have at the moment, but... getting perilously close, honestly. Someone who really dedicated themselves to taking "gmail inboxes" and writing a system to determine social security numbers from that could probably do pretty well. It wouldn't quite be just "fire some machine learning at it", but the system as a whole seems pretty feasible to me.
[1]: http://www.stevemorse.org/ssn/ssn.html . It isn't quite as bad as it seems for many of us, because we didn't all used to get SS numbers at birth, so my SS number does not correspond to my birth. But if you have my last 4 numbers and a handful of other bits of information about where I've lived, it's distressingly few bits between me and my identity getting stolen.