Short Answer: Scientifically, the size of the arena is irrelevant to the shooting performance though the nervousness of playing in front of a big crowd can negatively affect a player's shooting.
Long Answer: Statistically speaking, the stadium doesn't affect shooting by much (and this website makes that case). To prove this point, below are the shooting percentages in the three different types of stadiums during the 2009 and 2010 regionals.
Hockey arenas (nine games): 42.8 percent, 455-of-1064.
Traditional domes (nine games): 43.1 percent, 444-of-1030
Stadium setup (six games): 42.4 percent, 290-of-684.
Compare these numbers above to the average median shooting percentage between the 2009/10 and 2014/15 NCAA season (which is about 43%), you'll see that the arena doesn't affect the shooting.
Take it from Connecticut's coach Jim Calhoun, who really said it the best:
Every gym has a floor and two 10-foot baskets, and Reliant certainly qualified
But then in some cases, such as an under 20% 3pt shooting in a 2015 regional semi-final and 29.7% shooting half in another, why do teams have disastrous nights? There are always a number of factors that attribute to these horrible shooting nights (this website explains one). Whether it be the nervousness that comes with being in a big game with thousands of eyes on you or great, lock-down defense, bad nights can't be prevented.
Conclusion: Even if the NCAA somehow determines that big stadiums do affect shooting, they wouldn't move the games into the team's home or smaller arenas. This year, over 70 000 people watched the NCAA basketball finals in person, which resulted in millions for the NCAA and member institutions. In the end, with the big arenas, many more get to experience the thrill of the NCAA basketball game.