I was following in Real Time the BAT ICO, and my experience was a bit different. My parity node was up all the time receiving updates. Most of the BAT was distributed privately before the ICO started as you said. The 25% share is probably from BAT team themselves so the Cap was already near Max when ICO started. A really lot of people confused Gas with Gas Price, putting a lot of Gas on transactions a standard Gas Price. People who have correctly scheduled the transaction and put a Gas Price over 400G got mined and included on the limited available BATs. It looks like Team already have the majority of money from private investors and choose to sell not over ICO but on exchanges when the price was a lot higher.
RE: Crypto Currency ICO's, ETH Scams, and what your really need to know about BAT coin.