I think the AT&T Center has a couple of major differences from the normal way that teams extract stadium blood money from local and state governments.
First, the AT&T Center's public funding came -- as others have said above -- via a tax hike on hotels and rental cars, that put the burdens of the public part of the arena deal mostly on the backs of people who visit Bexar County rather than those who live in it. Since the tax was new, the County didn't really "lose" money on the deal.
The other major difference for Bexar County is that it retained usage rights to the building for periods of time at least and has the ability to realize revenue at least from the events that it hosts in the building. I'm sure there's a legitimate criticism that the County should have retained even more interest in the revenue generated by the building, but to the extent that Spurs Sports & Entertainment maintains separate smaller companies that handle various aspects of the building's operations (parking, for instance), farming the revenue out to the Spurs actually comes at the savings to the County of having to staff and finance those things.
Public funding of sports arenas has become a significant problem -- and, as bad as it is here, it's far worse with larger events run by ruthless en ies like FIFA and the IOC -- but I've thought that San Antonio/Bexar County handled the financing part of both the Alamodome and the AT&T Center about as well as any government has in building those sorts of projects.