There are a few different aspects to this.
Perhaps the Bible Belt has an attitude that gambling is a vice, as taught by the church. A lot of fans don't agree with this morality, and I suspect this is not a big driver for those who run the sports leagues.
Moreso, I think is the fear that someone in position to influence the outcome of sporting events, for personal profit, may gamble on a sporting event. Such as betting for / against one's own team, and then altering his / her approach to the game to achieve the desired outcome vis-a-vis the bet.
This latter argument has shades of grey. Some of the 1919 White Sox threw the World Series by making timely errors / ill-advised plays = VERY BAD.
Pete Rose admitted gambling, but contends he did not bet against the Reds.... But might he have altered his pitching rotation in order to create a more one-sided match-up on specific days, in order to facilitate winning bets placed on the Reds ? And by doing so, not optimized the Reds chances of winning as many games as possible ? Don't know ... depending on the facts, the Pete Rose case is debatable.
Neuheisel participated in a NCAA basketball pool. Clearly he had no intention or ability to corrupt the event. But with shades of grey possible in some scenarios, the sports leagues have in many cases taken blanket approaches to outlawing participation in gambling altogether.... therefore Neuheisel got tainted, as did Willie Mays for once endorsing a casino.
I can see the value of rules that prevent an incentive for altering the outcome of games. The rules need to be clearly laid out in advance, and well-publicized, and have clear lines drawn with regard to what is permitted and what is not In Neuheisel's case, the university seemed to jump to a very strict enforcement of a very general interpretation of the NCAA ban on gambling, and did not afford Rick Neuheisel due process, and that is why Neuheisel came out ahead when he threatend to sue.
So I think this has more to do with preventing corruption of the sport, rather than some church's view of morality.