When I turn on threat analysis and it gets to a depth of 24, it says Black wants to play Qe7 or Nh7 (after the latter, White's recommended move is again Kg2). I imagine the point of White's move is to prevent something from happening, maybe some attack along the semi-open f-file, but I don't really see any concrete moves for Black that would require White to play Kg2 with such urgency.
I realised as I finished typing up my question that if White wants to trade off the knight for a Bishop (thus retaining the bishop pair, which is good for reasons, probably) White can't just play 3.Ne5 immediately, because after 3...Bxe5 4. Bxe5 Black can play 4...c4!, and now threatens to fork the king and the bishop. For example, you could continue 5. Qxc4 Rc8 6.Qa6 Qc5+, after which White is still better, but then Black has counterplay and active pieces (e.g. Black can play ...R8d8 and have pressure along the d-file, and apparently ...Ng4 is playable as well, but I don't understand that line).