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> so however many turns it takes to promote that first pawn, that's as
fast as the game can possibly go, so I do see it as fast.

Well, so apparently in any game with promotions the promotions will
eventually become 'fast', no matter how deep the board, or how slow the
Pawns. So it doesn't really put a restriction on anything, when you said
before that 'promotions can only affect a game when they are fast'.

> And by "linear", I mean in that situation, there is nothing else you
can do. It has gone from game to puzzle once there is a guaranteed win that
a human expert can conceivably see.

But this is how all Chess-like games end: either in a mate-in-N checkmating
problem, or in an elementary end-game like KQK or KPK.

The point of my example is to show that even a small advance in games with
promotion becomes totally decisive, no matter how deep the board. A large
fraction of the games will reach a position which is drawn, but would be
won if one of the sides had just advanced one rank. So you cannot afford to
wait for the opponent to come towards you even if he is still very far
away, as you could in 16x120 Chief. Just letting him step one rank forward
(even if only his Pawns do it, or some of his Pawns) turns so many of the
possible endings of the game from draws to losses that it gives you a
significant disadvantage.

If you are in a position with a Pawn structure that would make a lost Pawn
ending, even if you have still pieces it gives you a significant
disadvantage, because you can no longer afford to trade the pieces. Many
winning strategies of the opponent could be based on this (putting you in a
position where you can only prevent the loss of more Pawns by trading), and
you cannot match them with the reverse threat.