I did that, and I actually got a better rate through the broker. BoA wanted 4.5% at the time, about 2 years ago. I was able to get a 3.25% FHA loan through my broker and 4% conventional if I wanted to. Where the bank came in better was willing to reduce the fees saving me 1-3 K from what I remember. But it was a higher interest rate so...
I chose FHA due to lower interest and minimum down payment. I was one of the last ones to get grandfathered in for no PMI after 5 years with minimum (3.5%) down. Now anything less than 10% carries PMI for the life of the FHA loan.
I purposely chose to put minimum down since my cash on cash return is higher and I have enough cash left to put 25% down on 3 more houses.
I guess I could shop around myself, and I did apply with a credit union as well, but their communication was so poor, I did not realize I was approved until I had already closed escrow. Using a broker seems to be for the convenience of one stop shopping for the best deal, yeah?