I'm not sure if the GP here was edited to be substantially different, because your response doesn't match what they are saying.
AT&T wasn't desperate, and I'm sure Jobs would have preferred to skip on exclusivity, since that really didn't serve iPhone.
"Desperate for more customers"? Well sure, but I'm sure Verizon was even more desperate for more customers considering they were so close to being number one (AT&T was number one at the time, by a slim margin). After all, Pepsi literally bought restaurants and forced them to serve Pepsi in order to force consumers to drink their sodas from the runner up spot.
I don't have any reference for this, but just on its face it would make sense that if Apple needed to sweeten the pot for a carrier to allow it, exclusivity was a way to do it, and it would have been in Apple's best interest to pick the carrier with the biggest market share at the time, which was AT&T.
EDIT: Others corrected me that at the time the deal was signed in 2005 and during the famous 2007 keynote, the partner was Cingular. Through a series of M&A Cingular became what we know of as AT&T, and by the time the phone launched in July 2007, it was AT&T who held the deal.
Interestingly,
> Cingular chief executive Stan Sigman signed a secretive deal with Apple in 2005 before seeing any designs or prototypes of the phone. Other carriers had been scared off by Apple's reputation for controlling every aspect of its products. But Cingular and AT&T leadership saw the partnership's promise.
In Jobs' keynote he referred to Cingular as the #1 carrier with 58 million subscribers. I think the logic above still makes sense given that.