The main thing that speeds the fix is the fact that they download the satellite positions over the data network. Satellite positions are embedded in the GPS signal, but they are transmitted very slowly. That is, I believe, the 50bps NAV signal he discusses on that page. It takes about 30 seconds to download it all over the GPS signal, while it can take <1s to grab it from your local cell tower. This is also the big reason why "warm starts" of standalone GPS receivers that have only been off for an hour or two are so much faster: the position data is good for a few hours, but after that it has to be downloaded again.
So cell phone GPS receivers have everything needed for standalone GPS, and position assistance from other sources as you mention, and the ability to load satellite position data from other sources. All this on a chip that probably costs less than $10 in bulk.
