Since many respected endgame books site that the most valuable piece should be activated first, the rook should be activated before the king, in general of course.

In this particular instance, black, in capturing the d-pawn, actually prevents his own king from getting access beyond the 2nd rank by taking away the d5 square and losing time. White's rook then dominates the black king, allowing the white king to walk up the board more or less unimpeded.

Do you see anything special with 34.Kg2? I feel it may allow the black king to move up and blockade the d-pawn himself.

After 34.Rc1 Rxd5 35.Rc4, Black's rook is tied to the c-pawn and cannot retreat or activate itself

After 34.Kg2 Ke7, Black's king can help out and cannot be easily imprisoned as in the game.

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Now, if White tries the 35.Rc1 maneuver, Black has 35...Kd6 freeing his rook and getting the king in play. Note also that White cannot easily bring his hing to the important central area, as Black has the option of playing ...f5 if White moves Kf3.

By the way, after 34.Rc1 Rxd5 35.Rc4, why doesn't Black try 35...Ke6 tryhing to activate his king? I think the answer would be 36.Re4+. If Black's king moves to d6 or d7 he blocks his rook and 37.Ra4 wins the a-pawn. If 36...Kf5, White penetrates immediately on e7, and 36...Kf7 37.Ra4 is essentially the same as the game.

The power of a rook on a6 imprisoning Black's king is a classic endgame theme. Here is one of the most famous examples: Rubinstein vs Lasker, 1909