This one's easy - no money. In spite of being the 6th largest city in the US, San Antonio is one of the least affluent. We have a very large population of brothers and sisters from our southern neighbor, many of whom have menial jobs. Most don't pay taxes, but we still have to provide medical, police, fire, and social services.
We have a great NBA team, but that's about all we can afford.
We have tried both AFL and MLS, but both failed here.
To get an NFL team we would have to build a stadium, and it would be expensive, because it would have to be large and need a roof for A/C - we don't get cold weather here (snowed in 1985). All of the new stadiums seat about 100,000 people and most of our population cannot afford season tickets.
I moved here from KC where we had the Chiefs. They were also marginal, in terms of business, but were able to pull in support from Omaha, Des Moines, Wichita, Topeka, Columbia, et cetera. We already have NFL teams in Dallas and Houston, so that's out.
We do have a minor league baseball team, but even those games are sporatically attended.
I'm afraid an NFL team would go broke or move quickly if located in SA.
There isn't enough fan base in SATX to support an NFL team. Mostly Cowboy fans down here. Plus, the new team would want a new area, the Alamodome wouldn't cut it. That's another topic all together. Good thing the city wasted our tax dollars on that well thought out plan.