Heading bug does not align with compass heading when HDG knob is pressed. Cessna 172 - G1000

No, heading mode has nothing to do with wind. It simply sets the direction you point in, nothing more.

This is important when flying IFR, as any vectors ATC give you are to be flown as instucted, without any wind correction from the pilot.

The fact that the sync button sets a value other than straight ahead is a bug.