You're omitting a fair amount of complexity here, I believe.

The "$350 in-branch offer" doesn't really even exist, though a few people have had some hit-or-miss success.

I actually track this stuff (goodness knows why), and the current reality, according to the card's dedicated thread, seems to be approximately as follows: "Since I could not get the in-branch offer on the phone last night, I went to the branch today during lunch hour. Initially I was told the $350 Annual Fee [offer] is only for Citigold [account holders]. I am not a Citigold member and do not want to be, so the guy checked with the manager and told me there is a way to get around and will call me in a couple days see if application is approved."

I sincerely wish everyone who tries this good luck, of course.

More to the point, there's also a no-fee offer out there for Private account holders, which is actually the offer that I took.

My advice? Unless you're living paycheck to paycheck, go ahead and flip $50K into a Citi Gold account arrangement (it doesn't have to be just in checking & savings) and get the "legitimate" $350 offer.

This then has the added benefit of churning out 11,700 Thank You Points per year for free, which are worth exactly $187.20, of which $70.20 are the "extra" value provided by the Prestige's 60% points bonus. This is assuming, obviously, that you fly AA every once in a while and so can take advantage of the ability to buy AA tickets with points.