What make you think that the existing program was a regular cab? One would hardly describe "NFL players can call cabs" as "an existing union program where they call a phone number for a car" without intending to mislead.
The barely-described prior program could be slower than calling a cab, if it's a national call-center that then picks a city-specific service, and doesn't draw on as many providers. Clearly there have been problems with it.
And if you're drunk and out past some contractual/implied curfew, talking to strangers is harder: there's some embarrassment involved. Pushing a button in an app depersonalizes the situation, helpfully so.
According to this article (http://www.thepostgame.com/features/201304/nfl-players-worry...) the major problem is privacy - players worried that the team management could find out that they'd been drinking (and to a lesser extent, cost, and perhaps to a greater extent, stupidity. Cost will be reduced with Uber, privacy should be less of a concern, and stupidity will remain unaffected).
