Why is the move 4.Nc3 rare in Bird's defence? After:

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1. e4 e5 2. Nf3 Nc6 3. Bb5 Nd4 4. Nc3


it looks like white is leading in development at the moment.
While it is true that at the moment white is leading in development, the move Nc3 interferes directly with white's plans. This is why it is seldom played. From now on I will severely criticize this move, but it is by no means a losing one or one that offers any immediate advantage to black.

In every opening, but this is particularly important in the Ruy-Lopez, the two main goals are to control the center and develop your pieces (remember, a piece is developed when it's doing something useful in the square it is. Sometimes a piece is better left on its starting square if you don't see a good place for it). To do so, a normal plan after Bb5 is to play c3 preparing d4, when white will achieve the perfect center (namely a pawn on d4 and another on e4) even if black captures on d4, as he will be able to recapture with a pawn. So, even if your knight is "developed" (my next point will be that the knight is probably not specially useful here), it prevents the pawn from reaching c3 and interferes with one of the two main goals of an opening.

The move has also a number of drawbacks that are conceptually maybe not so important as this one, but that also put people back from playing it. The most common is that after black plays d6, the natural square for his light squared bishop is g4 pinning the knight. If white is able to play Nbd2, he defends the pin and the queen is free to move without compromising the pawn structure, but if his knight is already on c3 the pin can be quite annoying. Another important one is that if the bishop is chased away via a6 and b5, sometimes he can hugely benefit from being placed on c2, defending the e4 pawn and having attacking prospects on the kingside if the pawn ever moves.

Finally, in this particular position, with this move you are giving up the bishop pair. It is true that you gain a temporary initiative, but with so few developed pìeces it is hard to get an attack going, specially when black is only a couple of moves away from castling. I do not think this initiative is something special, whereas the two bishops will bring a lot of pain after c6 and d5 (not necessarily right away, but that is a plan), opening up the position.

Summing up, while you are developing a piece, a considerable number of small concessions are being made, and this adds up to something that you may do not want out of the opening: you have poor coordination from the start, you may have lost the bishop pair and not even have any structural advantages for it (as in the Exchange Ruy-Lopez) and you must look for a new non-standard plan as you have blocked your c pawn.