Why is it necessary for black to defend the e5 pawn after 1. e4 e5 2. Nf3? What do you think of 1. e4 e5 2. Nf3 a6 3. Nxe5 d6? Now white should move his knight to c4, d3 or f3 and black can develop more pieces.
Otherwise, the real problem for Black is not the knight on e5. It's the fact that you're giving away a center pawn without any compensation whatsoever. If you want to develop pieces, you're even better off playing 2... Bc5 than 2... a6.
The purpose of protecting e5--or counter-attacking by Nf6--is to prevent the lose of material.
After 1. e4 e5 2. Nf3 a6 3. Nxe5 d6, white is up a pawn and has equal development. Black doesn't have enough compensation for the sacrificed pawn. 3... Qe7 regains the pawn, as white can't protect both e5 and e4, but the resulting position will leave black's queen exposed to attack from white's minor pieces and gives white a greater advantage.
To keep the game equal, black must protect the pawn with Nc6 or d6 or counter-attack with Nf6. Protecting e5 with f6 loses to a famous chess trap(https://en.wikipedia.org/wiki/Damiano_Defence).
The idea of offering a pawn for a lead in development is called a "gambit."
Gambits are mostly played by White, who has the advantage of the first move, and who gets the advantage of a second or even third move for the sacrificed pawn. This can make it worth doing for White.
It is not worth doing for Black, because he would have only "switched colors," gotten White's normal first move advantage, but with a pawn down.
Your proposed line 1.e4 e5 2.Nf3 a6? 3.Nxe5 d6 simply loses a pawn for nothing. Black gets no compensation in the form of development, because time has been lost by the pointless move 2...a6.
The two main replies to 2.Nf3, 2...Nc6 and 2...Nf6, both have the advantage of developing a piece and not losing a pawn.
Only defend the pawn because you coincidentally develop your pieces as well.
Playing 2...d6 to defend the pawn because you're SCARED it will get taken is a bad attitude.
Even 2...f5 is a super, hyper-aggressive move that I'd play instead of the timid and mediocre 2...d6.
But you see the attitude? If you decide to play 2...d6 with a solid strategy, and you REALLY know what you're doing, then it's playable. See Philidor, Hanham.
Playing 2...Nc6 develops a piece at the same time.
Even 2...Qe7 can be played, but it's a bad strategy. Good night, King's bishop. No pun intended.
P.s. please forgive the caps. I'm on my phone and don't know how to italicize.