Aircraft display systems for deployment onboard rotorcraft or other aircraft are provided, as are methods carried-out by an aircraft display system. In an embodiment, the aircraft display system includes a controller operably coupled to at least one cockpit display device. The controller is configured to generate a Primary Flight Display (PFD) and Horizontal Situation Indicator (HSI) graphics on the cockpit display device. The controller selectively switches PFD operation between a heading-centered display mode and at least a first non-heading-centered display mode. The HSI graphics are produced to include one or more movable HSI symbols identifying a current aircraft heading when the PFD operates in the first non-heading-centered display mode. In certain embodiments, the controller may also selectively generate visual alerts on the PFD by altering the appearance of the HSI graphics when an angle between the current track and the current heading of the aircraft exceeds an angular threshold.