"The 3rd and 6th ranks have pathways that only pawns may enter."
My answer was deleted, I assume because Mr. Towers failed to understand the reasoning and took it as gibberish. I guess I need to elaborate and defend it.
Another answer I have seen is to say the pawns are lightweight militia units marching double time and the other heavy units cannot catch them but other pawns can. Pawns, interestingly, only have this burst of speed on their first go.
This tries to make sense of the game and is one interpretation of the more general answer I provided.
Why can a rook, bishop, king, knight, and pawn not duplicate one another's movements?
Because this is what the rules say?
Not technically wrong, just entirely unsatisfying and missing the relevancy of the question.
Moving from point A to point B is not done by the same path for each piece. Which can be interpretted many ways such as having different vehicles, as the rook(chariot)/knight(horse)/bishop(elephant), or political influence, such as the queen(vizier)/bishop(clergy) etc., or it can be seen as a more abstract mathematical concept underlying the game itself, which takes the game back to its earliest roots.
Look at Chinese or Congo chess or Alice or 3D chess to help wrap your mind around the concept. For example, in Alice or 3D chess, pieces literally leave the board and travel a way before returning. In Congo or Jungle chess, pathways are physically laid out that characterize how a piece can move. Such concepts have been around a while.
While there is no historical manifesto for why the pawn can only capture en passant, what made sense back then possibly makes sense today; still being the same game.
The paths followed by the orthodox chess pieces, exist on hidden layers of the chess board while the pieces stay tethered to the base layer following moves, where they can be displace one another, except the pawn, during its double-move, actually removes itself from play, skips the opposing player's move, moves again, and returns itself into play, all in a single move.
A queen has access to the layers or paths of the rook, king, and bishop. The amazon/dragon piece from Maharanja and the Sepoys has the additional path of the knight added as well. Even still, the pawn remains uniquely separate from all these other pieces.
You can't expect a monkey to follow a mole underground or a fish to climb trees, after all they use different modes of travel, taking longer to reach different locations by different paths and perhaps having no way to reach certain locations at all.
Only another pawn, which moves along the same hidden path, is able to intercept another pawn making this special double move. While such a route of movement is inaccessible to other pieces, this is a dimension other pawns are familiar with and they can still reach through it in order to displace a pawn during its double move.
Pawns are basically time travelers, and only another time traveler can catch a time traveler.
I appreciate all other efforts to justify the reason, even those that are clearly wrong, and I don't see why any of them should be deleted; after all, the answers are voted on.
There are paths on rows 3 and 6 that allow the double move and only other pawns have access to those paths.