*While the social engineering concern is a valid one, I suspect that Google's original reasons for not providing support were not opsec related.
Google's 2FA system combined with their lack of support terrifies me as a security-literate user. Here's one example: I was traveling and signed into Google from a new location. Google prompted me to verify myself via two-factor auth, and the only method they allowed me to verify by was by opening up the Google app on my iPhone and by confirming the provided number. I tapped "try another way", and gone was the option to verify via authenticator app.
I'm lucky that I had my phone on hand but I was dumbstruck. What if I lost my phone? I'd be screwed, locked out of my account with no way to fix it, even after following the best security practice of enabling 2FA via authenticator app, because Google took it upon themselves to say "screw your choice of security, open our app on the phone" (also, thereby coercing me to link my two devices to my ip address/location for analytics/targeting reasons, I'm guessing).
I could have done everything right and still gotten locked out of my account.