Consumer GPS chips are specifically nerfed for using them in rockets; they give erroneous readings on purpose if altitude is above a certain height and/or if speeds exceed a certain speed. That’s likely why the mid-course correction software uses other methods.
Chinese GPS chips dont have those restrictions.
I even have 1 that can remove up to 8 active jamming signals.
Gotta love what you can buy for $20
What you are likely thinking of is the "selective availability" system, which intentionally provided slightly inaccurate data to civilian clients, while military receivers could decrypt the most accurate info. But this has not been used for many years now.
Other than that, GPS is a one-way system, it does not know you exist, how fast your receiver is moving or "give" different information to one client vs another.
Even if it did, this is essentially a toy and moving slower and lower than a general aviation plane.
It uses accelerometers and other sensors because they can be sampled and integrated hundreds of times a second. The $5 gps module is 9600 baud serial and provides one update/second (or maybe 5/sec depending on which part number you pick).
The restrictions on GPS prevent ballistic missiles, not MANPADs. Typical limits are 515 m/s and 18,000 meters (try using your phone's GPS on a commercial flight, it works fine near a window). Update rate is probably the biggest issue with GPS and MANPADs.