Does it make sense for White to play as "Black" when he's lost the initiative? In this game, White appears to have lost the initiative against the Nimzo Indian defense as early as the fourth move, and certainly by the eighth move.

If White's next four moves were 1) e3, 2) Nf3, 3) Bishop moves and 4) O-O, White would get a solid defensive position that is characteristic of Black's play. Under the circumstances, does it make sense for White to "play Black" in this way if he wants to minimize his chances of losing?

(In the actual game, White adopted a more aggressive policy and succumbed to Black's superior development.)

One other feature of note. White was rated about 50 points weaker than Black, even though they were both Masters (2400 vs. 2450). Would it make sense for White to "play Black" if he were 50 points weaker, and play more aggressively if he were 50 points stronger?
This is actually quite common behaviour in the Nimzo-Indian defense. Black trades the bishop pair for either an advantageous pawn structure (e.g. in the Hübner variation) or a lead in development (like this game).

In the latter case, a successful strategy for White might be to neutralize Black's lead in development (which corresponds to your "play Black"), exchange some pieces and take advantage of his bishop pair in the late middlegame or the endgame. However, this style of play might not suit everybody.