> it seems to me you are essentially saying that any game with promotion
would have pretty much the same white first move advantage, if I understand
you correctly. At least, it appears to me that it follows from everything
you've said.

Well, nearly so. There could be additional advantages on top of being
closer to promotion that can be achieved an just a few moves, and they
would add to the first-move advantage. Like 'developing' pieces, e.g. by
increasing their general mobility, or specifically directing them against a
weak spot in the opponent's setup.

Another caveat is that the promoting pieces should be short-range. In a
game where only Rooks promote (to Dragons, say, like in Shogi), advancing a
bit would not be helpful. They could promote in a single move wherever they are; this only depends on whether the file is open, not on their location in the file.

In Chief there is no promotion, so advance is not likely to be worth much,
as the initial position is already quite open. (I could imagine that even in
Shatranj, with hardly significant promotion, advancing all pawns of one
side by one rank would still be helpful to that side, because he can use
the open rank to laterally move his pieces, especially Rooks.) But IMO
there still should be an advantage to having the move, as moves can be used
to set up an attack formation. In particular by concentrating your pieces
for attacking on a single point along the lines of the opponent formation,
which is laterally spread over the entire board.

Spreading is bad strategy if there is nothing worth defending. Against a
concentrated opponent, part of your army will be outgunned, and there is no
compensation where your army outguns the opponent, because there is no
opponent to kill there. (And taking the undefended area does not provide
any gain, as empty area isn't worth anything in absence of promotion.) So
part of your power is wasted. This is why I think of the Chief array as
'undeveloped', so that sitting idle is a bad idea, and will quickly lead
to a lost position if you do it too long. (And the opponent uses the
opportunity to contract his forces in the central 6 files of the board
laterally, say).