Our school switched from ASB to the School's Foundation (read: 501c3) a couple years ago. Luckily for us, the check turnaround time is about a week and most of it is reimbursement based.
If I give any advice it is this: as the lead mentor and #1 contact between the school, foundation, and the rest of the team, never underestimate the value of having a good relationship with your foundation and school contacts.
A couple years ago, our team had a lot of worries about switching to the Foundation. These were based on a couple things, such as the Foundation didn't want to separate our funds into an account, or if they did they wanted to take 25% towards general operating costs. It sounded very weird to us and we were anxious about taking the leap.
It wasn't until I sat down with the people running the Foundation and talked it out with them that I got the complete picture. Yes, some of the fears were true. If we switched to the Foundation, legally the money that got donated could be used by other science and engineering areas of the school.
But, and this is a big but, by us having a relationship with the school, that doesn't matter as much anymore. The people directing the money know what type of funding we bring in, and we know that we will be taken care. In two years we've never had an expense denied and they've been thrilled to help us do incredible fundraising things (making a robotics camp that brought in 82k revenue this summer) that might have been impossible otherwise. We've actually worked with them to use money that we didn't earn to improve our workshop and build out our program. Its been a huge blessing.
I think for some teams, forming their own 501c3 will be the best move. However, I would highly encourage you to really look at your history with your school foundation, sit down talk with the people in charge, and try to see if you can make it work. Often times in these situations
bad blood or miscommunication can form that can leave both sides with bad tastes in their mouth. But, given that both sides are interested in educating students, I firmly believe that it is worth a try to make it work.
Also, less paperwork and things to worry about. Running an FRC team is hard enough.