Endgame: Rook vs 1 (or 2) pawns

Evening folks! Recently I started studying Silman's complete endgame course and saw something interesting on page 77. A match between Leko-Kramnik where the following position arose:

Even though black was a rook up, he could not win the game. Important to note that the rook can only win the game if the king is there to assist with the capture.

The above position was quite an eye opener for me, as I was under the impression that a rook vs pawn ending should be an easy win for the side with the rook!

In a recent game I managed to reach something similar to the Lucena position and thought that I had a guaranteed win (didn't know the Leko-Kramnik game then) However after the promotion and rook capture that followed, I suddenly realised that I may in fact be losing! The difference here of course that there are two pawns and the rook can only be sacrificed to forcibly take one of them, if the situation really turns dire. In reality it wasn't so bad and only draw should result from this position. But if either side is not careful then the game could result in a loss/win as well!

The following transpired (spoiler: I got lucky )

I've analyzed this a bit and to me it looks like e4 resulted in a loss because:
1) It removes protection from the d3 square, allowing the white king to get in front of the pawns via d3-e2
2) e4 opens the black king up to checks from the side
3) the advanced pawn is closer to the white king, making it easier to block/capture.

The above points also servers a greater cause: It is very good for the side with the rook to get his king in front of the opposing pawns (and his opponent should try to prevent this). This helps to stop immediate promotion threats and pawn advances as well as helping the rook to eventually capture the pawns.