If anything, the tail number does not matter nearly as much. A plane with auto land presumably already has ADSb out (almost certainly 1090ES), is squawking 7700, and is probably already IFR anyway. As in this situation, the controllers knew well in advance they had an emergency inbound and who it was. At an uncontrolled field, I need something to tag (robotic "bravo-romeo" is plenty) and a relative position. Bonus if it does the math and predicts landing time, which it does.

Frankly, it should know (like I have to) if it's going to auto land at a towered field or uncontrolled, and adjust as necessary to those circumstances.

I’m not sure I agree. Not sure I disagree, either. If I’m another pilot in the air when this occurs, it feels like the most important things for me to know are (1) stay the hell away from the runway, and the announced approach, for a while; (2) only a single aircraft is doing an emergency autoland currently; (3) assume that the aircraft will need medical response while on runway (no auto-taxi) so if I was planning on landing in the next half hour or so, go to alternate. (1) and (3) are well covered, but (2) is subtle — /today/, the chance of two aircraft doing an emergency autoland at the same field at the same time is negligible, but it’s still something both I and the system designers need to think about.