I took issue with calling out the airport, multiple times in full phonetics, both at the beginning and the end of the transmission. All other callsigns, perfectly reasonable.

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.

At an untowered field, saying the airport name at the beginning and end of each transmission is standard phraseology.