Why Don't Pros Push Pawn to E5 Against King's Indian Defense?

Since you're asking such a question I don't think you'll be sub-1000 for long.
Sometimes e5 can be good, but usually it's a bad idea since it's best to maintain the tension. Also ...e5 is common in KIDs so white doesn't have an opportunity and even after exd4-Nxd4 the e4 pawn is quite well restrained. This is because, like some Pirc/Modern lines such an advance if it were sound would be decisive. Thus, a lot of basic theory revolves around preventing it. Even as far back as Nimzowitsch restraining pawns was known.
As for the very specific position you posted, well, Godlike already covered it quite well.

What if white push pawn after black castle?
usually its premature, black can just play ne8 and undermine the big center you have build up, wity moves like dxe5 and/or c5.
in general, white wants to maintain central tension in KID unless black has made some crucial error or has done a move that forces a momentous choice (like e5 or c5 by black). White playing e5 is only good in very specific circumstances. Focus on developing your pieces instead, in these kind of hypermodern defenses, the pawn stampede rarely works early on.
e5 is anti positional: it plays the pawn for a second time, it does not develop any piece into play and it weakens the white center, where the pawns are strongest shoulder to shoulder d4-e4.
Here is an example of black punishing white for playing e5:
https://www.chessgames.com/perl/chessgame?gid=1008397
I'm basically a beginner, but though I understand and like the KID and its concepts and lines, I don't play it against 700-1000 level players I'm paired with. They ALWAYS push the pawn to E5. This level player always pushes pawns if there is any question. This means that even though you get a better position after the pawn exchange, the queen exchange, and Ng4, it guarantees that white won't castle kingside, a queenless middle game, and having to make your kingside attack without your queen. When people make tutorials for beginners about the KID, they don't seem to understand that it needs to be 90% about middle game concepts without same side castling and no queens.
Below are the typical intro. moves to the King's Indian Defense against WHITE's D4 opening.
In online games here at Chess.com, WHITE frequently pushes his/her pawn to E5 attacking BLACK's Knight on F6, which I find to be annoying (forcing me to move it).
But, it seems GMs/professional players (from videos of games I've seen online) don't do this as WHITE. Why is that? They seem to take other lines. I've always been curious what they WOULD do or how they would handle that pawn to E5?
Is there something theoretically wrong with that move that BLACK can punish? Thanks a bunch!