iPhone EXIF/GPS Timestamp - Appears to be based on local settings?
Am I correct in this assertion that the GPS timestamp doesn't come
  from the GPS at all? Or am I missing something?


Your experiment definitely makes it look like that!

This very detailed web page Exif Schema for Exif-specific Properties says:


  GPS Time Stamp: GPS tag 29 (date), 0x1D, and, and GPS tag 7 (time), 0x07. Time stamp of GPS data, in Coordinated Universal
  Time. Note: The GPSDateStamp tag is new in EXIF 2.2. The GPS
  timestamp in EXIF 2.1 does not include a date. If not present, the
  date component for the XMP should be taken from exif:DateTimeOriginal,
  or if that is also lacking from exif:DateTimeDigitized. If no date is
  available, do not write exif:GPSTimeStamp to XMP.


It looks like what's happening is the File Modification Date/Time shows the local date &amp; time according the phone's clock, in the phone's time zone.

This is then converted to Coordinated Universal Time ("UTC"), and saved as the GPS Time Stamp.