Going to the "wrong" Academy with the intent to cross-commission is stupid. Here's why!
At the start, it's a very small number that get approved. At most, one or MAYBE two per service. The reason why it's so few isn't just because it doesn't make sense to go to Navy when you want to be in the AF (or whatever). The Navy has to let you go. They've sunk a lot of money training you to be a Navy or Marine Officer and they're reluctant to just let all that money go so you can do something else. You need a very good reason. They don't want people who just hate the Naval Academy and assume the grass is greener in AF-land.
A girl in my company from 2011 just cross commissioned Coast Guard. She had other reasons as well (probably even more than I'm aware of), but she had spent a semester at USCGA and that helped her case. There was a USMA cadet in my platoon at Leatherneck last block and he said there they base it off of a series of criteria including family history and continued interest (along with some other stuff I frankly don't remember), and there were still no guarantees. Being a strong performer helps.
Another reason it's a bad idea is you have to be okay with not necessarily getting the exact job you want. For example, if you're at Navy and decide you want to be in the Army because you want to drive tanks so you try to cross commission, you would have to be okay with being in the Army and not being an Armor officer. At least here, you also might not find out whether or not you're selected until very late in the game. The girl in my company who went Coast Guard didn't find out until April that her request to do so was approved. Otherwise, she would've gone SWO.