No, an autopilot doesn't fly a heading
I will ignore most of your pompous rant, Mr "KG". But the above is actually how an AP works. It holds a
heading, which is then continually adjusted according to the HSI deviation bar error (taking the simple non-GPSS NAV case, with a GPS or VOR source).
The basic "heading" is taken from the course pointer setting (if a mechanical HSI; the pilot turns the CP to the new track at each WP), or from an EHSI CP setting which is automatically slewed to the current GPS track at each WP.
A GA AP doesn't fly asymptotic curves
You can get more intelligent behaviour with GPSS, whereby the GPS directly controls the AP, bypassing the HSI. This can give you accurate fly-by WP behaviour. The GFC700 should have this in all G1000 installations.
He clearly had little regard for following the regulations as the report shows he busted so many.
Well, sure, but busting regs doesn't automatically get you killed. I still wonder why the investigators reckon he was flying on the AP when it seems obvious he was flying by hand (and not doing too great a job, especially at the very end).
I don't take the position that anybody busting regs deserves to simply die. If one could find out what was really going on, one might take something home from it.