Earlier quoted context omitted.
The fast TTFF on a typical modern device e.g. your phone is because the device has the Internet, and so it can obtain all the information it needs from the Internet up front, it isn't magically brute forcing everything needed, that's not practical at all. The 12.5 minutes includes a rough multi-week almanac which you could perhaps brute force given available compute and receive capability (original GPS receivers have…
Yes, I forgot to mention downloading the orbital parameters over the network! Thanks for mentioning this as well. In this case, I was meaning to refer to brute-forcing the Doppler-shifts and PRN phases of each satellite, not the orbital parameters themselves. The project in the OP is able to get a position fix in less than a minute because, if the subframe timings are convenient, you can retrieve the necessary epheme…
Most decent modern-ish receivers tend to have pretty speedy aquisition time without any assistance data. For example, the reasonably ancient GPS running watch I use can usually get a GPS lock in a couple of minutes from cold with no internet access (in a wrist-sized device running on battery!), and even the two decade old SiRFstarIII chipset is specced to have a sub-minute cold start time without assistance and much shorter with - though I think that chipset was pretty advanced for the time.