For example, the S-TEC 55X will continue to fly the airplane in
GPSS mode if the PFD or ADAHARS fails, but heading mode would be
unavailable due to the lack of heading input. If the AP lost heading
data during HDG mode operation, it would begin a large circling
maneuver. If it was in GPSS mode at the time of the failure, it would
continue operating seamlessly.


I realize you probably left this out on purpose to be brief, but technically
this is only true if GPS1 was already driving the autopilot or if GPS1 and
GPS2 have identical flight plans loaded and activated. It's not true if
GPS2 is flying the plane and GPS1 is inop or has a different flight plan
active. The Stec 55X will also still work in altitude hold or vertical
speed modes because its uses its own static pressure sensor separate from
the EFIS. These are the kinds of details autopilot users ought to know
cold.