I would say it's more of a math thing. GPS uses trilateration to calculate location.
You need four satellites to get a proper position and my guess is that GPS units don't show anything until the minimum requirements have been met.
So, that's an all or none affair, rather than piecemeal like you are suggesting. I think that's especially true since one of the big components of calculating position is time.
Something along the lines of "you need all the variables in the formula in order to spit out the result."
You may be able to find the actual formulas and math on gps.gov as well.
I know that smartphones can speed up position locking by using tower locations, but I don't know how that works with GPS in terms of hardware or software. It sounds like tinlyx may have a better grasp of that than I do.
Other links that may be useful:
Addition: I found some more info that may be useful, although I'm not certain it fully answers your comment.
The following is an excerpt from this article:
When you measure the distance to four located satellites, you can draw four spheres that all intersect at one point. Three spheres will
intersect even if your numbers are way off, but four spheres will not
intersect at one point if you've measured incorrectly. Since the
receiver makes all its distance measurements using its own built-in
clock, the distances will all be proportionally incorrect.
The receiver can easily calculate the necessary adjustment that will cause the four spheres to intersect at one point. Based on this, it
resets its clock to be in sync with the satellite's atomic clock. The
receiver does this constantly whenever it's on, which means it is
nearly as accurate as the expensive atomic clocks in the satellites.
In order for the distance information to be of any use, the receiver also has to know where the satellites actually are. This isn't
particularly difficult because the satellites travel in very high and
predictable orbits. The GPS receiver simply stores an almanac that
tells it where every satellite should be at any given time. Things
like the pull of the moon and the sun do change the satellites' orbits
very slightly, but the Department of Defense constantly monitors their
exact positions and transmits any adjustments to all GPS receivers as
part of the satellites' signals.
This description suggests to me that GPS receivers receive an initial almanac showing satellite locations, but then go through an additional series of checks to ensure their locations are correct, so that any time and distance calculations are accurate. This would explain why you can see satellite positions, but not your location yet. The receiver has to wait until its internal clock matches the satellites before any calculations are made.
I also know that initial satellite almanacs can have some satellites in the wrong location, and only after receiving info from each satellite does the receiver know to update their positions before calculating distances and time. The next page of the article talks more about how receivers handle errors in location and time.