Two rooks apiece is quite a lot. I'm sure the king advances were only possible because of how many pawns were left on the board. In open positions (say 4 pawns apiece or less) where there are still a non-zero number of heavy pieces, you'd only want to move him out when there is 1 rook apiece. (Heavy pieces are the queen and rooks).
For example in a Queen vs Queen endgame, sometimes you have some tremendous trumps (centralized queen, passed pawn) but your king is too exposed to escape the perpetual check draw. I saw a game like this with Keres vs... I forgot... but it was in the Zurich 1953 tournament book.
Other than "in the endgame" the other general saying would be "when it's safe" i.e. when there are no mate threats. In the GM game, I'm sure it was impossible for the opponent's pieces to coordinate an attack on the king.
When is a good time to do this? You'll say, "Well, when the end game begins."
But when is that? I just watched a top game where both sides had 6 pawns, 2 rooks and a bishop or knight AND they headed toward the center with their Kings.
Was that premature? Is the answer, it's okay, so long as none of your pieces will BE beHIND your king and/or none of opponents pieces are likely to get behind your king? If the opponent has a bishop, it seems dangerous to move the king too early.
I don't know. What do you think? When's a good time to move your king toward the center?