Is there Anything Wrong with 2. Ne2?

Ive been analyzing this opening for days, and cant find anything wrong with it, in any variation. I have been playing games on Chess.com withit, and i have done pretty well with it. Why isnt it played? The main line:

Here are some lines I have produced

Even though he's down a full pawn?

A trap that may fool beginners

I think ItsEoin means this:

I'd actually meant exf Nxf Nxe but partway through I realized that, like you say, that line drops a Knight. My bad. 

see i dont see any game in my life that played ne2 this is bad really bad go for nc3 or nf3 but anyways the main moves after ne2 is bc5-d4-

The "trap" reminds me of a line I've won with a half dozen times in the Vienna.  In fact if you want to play f4 in the open game 1. e4 e5 but don't want to play the King's Gambit, I recommend this opening to you instead, where you will still have the option of the active and customary Nf3 rather than Ne2?!

Ne2 is allegedly called the Alapin opening and was featured in S.O.S. opening surprises vol. 4.  Why play 1.e4 opening lines for your queen and bishop only to close them again with 2. Ne2 tho. I lost a game against my cousin using this opening.

@ pfren: I realize that the "trap" is not difficult to avoid, surely not at the master level. But as a rated 1350 player, it works quite well on my opponents. I surely wouldnt expect an international master like you to try this on one of your opponents. This opening should probably only be used by lower level players like I to attempt an early grab at the center with the F and possibly D pawns.

Ex.

Although Black obviously made some serious mistakes, this kind of thing happens more often than you would think in my games.

btw, white is expected to castle queenside.