I applied to The Leadership Alliance last year, and as far as I'm aware they do forward all applications to the schools (assuming they receive your transcript by the deadline). I did find out that each school sees all the other programs you apply to and your rankings for them (Not necessarily a big deal, but it did surprise me a bit). I would suggest using their application as it does allow you to apply to three programs with one application.
As to whether you should apply through the school directly, that depends on the institution. Some schools actually want you to do both (Columbia is one, I think), but I'm not sure. You can always send a quick email to the summer coordinator to ask for their advice.
Bonus of applying to Leadership Alliance is you get to present at their symposium. It's a great place to meet students from other programs/network and get some research presenting experience.
