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 ephemeris parameters from the subframes in that span (and down to as little as 18 seconds in ideal conditions, if my back-of-the-napkin is right).
Yeah, the key reason this enables so much faster time to first fix is that the precise ephemeris parameters are transmitted much more often than the full almanac, but only from the satellite they apply to whereas each satellite broadcasts the entire almanac covering the whole constellation. If I'm understanding the info out there correctly, every transmission of the ephemeris data comes with only 1/25th of the almanac.
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.
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 ephemeris parameters from the subframes in that span (and down to as little as 18 seconds in ideal conditions, if my back-of-the-napkin is right).