What makes a game with promotion so different is that there exist multiple
strategic objectives. Otherwise there is no other possibility than to go
for the King, which makes it comparatively easy to set up a STATIC defense.
With promotion the attacking side has a choice between going for the King
and going for promotion. So if I allocate all my material to defending the
King, so that he could not possibly achieve anything there even when he
would direct all his material against it, he simply switches to promoting a
few Pawns, after which he will be able to crack my King fortress due to
huge material superiority over my best defense.

With multiple strategic objectives the game always remains a race, between
the attacker redirecting his forces to a new objective, and the defender
repositioning his pieces to fence of that new attack. This is why center
control is so important in Chess: it makes it easier to switch objective
when you are close to everything, be it for the attack or for the defense.

In a race, free moves are always valuable. And it is not clear at all that
extra moves are less helpful to win the race if your pieces are short
range, provided the opponent's pieces are too. In fact I would think the
opposite. For a King it is much more important to be in the center (when it
gets safe to do so, in the end-game) than for a Queen or Rook.

Note that this is completely different in Charge of the Light Brigade. This
is not a race at all; if it were the Queens would always win. But the
Queens cannot achieve anything against the numeric majority of the Knights,
no matter how many extra moves you would give them. So the Knights are not
in a hurry at all, and slowly but surely execute their own, unstoppable
plan to gang up on the enemy King.

Like I said, I am not convinced Chieftain Chess has no first-move advantage. Despite the absence of promotion. It could be that absence of promotion would allow building a static fortress that could withstand any attack, (after which there would be no advantage in having the move), but it is clear that the initial position is not such a fortress. In fact it seems an awfully poor position, because your forces are widely dispersed. If you would be so foolish as to allow the opponent to rearrange his pieces on his own half of the board before engaging you in tactics, you would be totally crushed. (E.g. I would contract my entire army to the right-most 1/3 of the board, and totally crush you there making use of my number-of-moves-per-turn advantage before you can get your distant Chieftains to the rescue. Remember that even if all your pieces are defended, when I capture 4 in the same turn, you can only recapture 2 if half your Chieftains are out of range.)