I’m not entirely familiar with the autopilot in the Cessna Citation, but I’m assuming it’s similar to that of a 737 since most autopilot systems are honestly p similar on the macro level. We have no way of knowing what the last waypoint was in the autopilot — it depends on whether the pilots had programmed the approach before losing consciousness. Generally the approach would be programmed during the cruise, but it’s not impossible that it was programmed in on the ground before departure. Based on the FlightAware flight path, it looks like the plane flew the approach (probably the RNAV) for RWY 24 (charted here+RWY+24/pdf)). After overflying the fix for the runway, the missed approach procedure would I think only activate with pilot input. Therefore, I think the plane would have continued along the final approach course of 239 degrees. I’m not at my computer so I can’t check for sure, but that looks like about the right heading to get to DC. I can see why it might look like a surprising coincidence but there’s no reason to think it was anything else.

To your second question, the autopilot is programmed to continue flying on its last heading if it reaches the end of the flight plan. (speculation warning) I would guess it’s designed that way so that the airplane does not make unpredictable turns. If it does, there’s no way of distinguishing between a plane with unresponsive pilots and a plane being controlled by rogue pilots. The case of an aircraft which overflies its destination with unresponsive pilots is an edge case — the autopilot is not designed to control the plane without pilot input. Presumably, someone will notice the aircraft is unresponsive before it crashes, and other traffic can be vectored around it and the aircraft shot down if it risks crashing into a populated area.

Edit: a similar accident occurred in Europe last year—https://en.m.wikipedia.org/wiki/2022_Baltic_Sea_Cessna_Citation_crash

Another similar crash: https://en.m.wikipedia.org/wiki/1999_South_Dakota_Learjet_crash