I think for most people it's because when religious programs offer "help", it's done with an ulterior motive. Your free meal comes with a mandatory sermon or some other way of making you feel worse about yourself than you already do. It doesn't take calculus to understand why most of us feel uncomfortable around religious people who literally believe that they are better than you because of their faith. I definitely try to find secular alternatives when looking for a way to help the poor as well.

To answer your hypothetical scenario, I would say if their methods were that much more effective then I would not have a problem with employing those methods with another more sane group of people. It's highly doubtful that such methods would be a result of religious beliefs and more likely linked to better financing and access to resources.