Why is the Cozio defense in the Ruy Lopez not played more often and what's white's best try to defeat it? The Cozio defense arises after these moves:

[FEN ""]

1.e4 e5 2.Nf3 Nc6 3.Bb5 Nge7


At first, this looks like just another dubious opening, but I found it surprisingly hard to prove an advantage with white. 

[FEN ""]

1. e4 e5 2. Nf3 Nc6 3. Bb5 Nge7 4. O-O g6 5. c3 Bg7 6. d4 exd4 7. cxd4 d5 8. exd5 Nxd5 9. Re1+ Be6 10. Bg5 Qd6 11. Nbd2 O-O


It is not clear how white should proceed. There are many options, but black has several different ideas and can quite easily general counter-play. Going 12. Ne4 Qb4 13.Bxc6 bxc6 seems like the most principled approach, but black is just fine in these lines. He can drum up quite interesting play judging from the grandmaster games I looked at. This line:

[FEN ""]

1. e4 e5 2. Nf3 Nc6 3. Bb5 Nge7 4. Nc3 g6 5. d4 exd4 6. Nd5 Bg7 7. Bg5 h6 8. Bf6 Bxf6 9. Nxf6+ Kf8 10. O-O 


is given by some FIDE-Master on YouTube, but I'm also not very convinced. I just fail to see white's advantage in this position. So, I have two questions:


Why is the Cozio defense played so rarely?
What is, in your opinion, the best way to secure an advantage against it as white?
The major reason is that it is played so rarely is that if white does not comply and take on c6, the Ne7 is poorly placed blocking the Bf8, and if the knight goes to g6, that just does not look like a good square either. In addition, it does not really help get in d5 since white gets there first with d4, or it just loses a pawn. Since that is not possible, the Bf8 must go to g7 where it just seems out of place with what most Spanish players are used to, and prefer. Also, most of the lines are still slightly better for white, even if they are subtle in nature.

There are three moves that are played on move four with a very high frequency: 0-0, c3, and Nc3. Nc3 seems like the best to me since it thinks very prophylactically, and stops d5 for a long time. In addition, the winning percentages for each move are 53.1%, 56.8%, and 62.6% respectively pre the Mega 2020 database.

I tend to like double e-pawn positions that white has more space because black has a d6 pawn. You can then eventually post the Nc3 on d5, cramping black further. I am going to propose that you play 4.Nc3, and aim for such a position since the plans for white are aggressive, and clear. In addition, I like positions with more space, in general, since I think that it makes it harder on your opponent in a practical game.

Despite my recommendation, I am going to add in some of the other lines so you can look at them, and decide if you prefer any of them. The other lines are also quite good, but they are also much more nuanced, and require very patient, and precise, positional play, compared to the main 4.Nc3 line I am proposing, which involves either a direct attack, or bringing the rooks to the center, and slowly advancing the kingside pawns.

 [FEN ""]

 1. e4 e5 2. Nf3 Nc6 3. Bb5 Nge7 4. Nc3 (4. c3 g6 5. d4 exd4 6. cxd4 d5 7. exd5 Nxd5 8. O-O Bg7 9. Re1+ Be6 10. Bg5 Qd6 {Transposes to 4.0-0.}) (4. O-O g6 5. c3 Bg7 6. d4 exd4 7. cxd4 d5 8. exd5 Nxd5 9. Re1+ Be6 10. Bg5 Qd6 11. Nbd2 O-O 12. Ne4 Qb4 13. Bxc6 bxc6 14. Qc1 Rfe8 15. Bd2 Qb6 16. Nc5 Bf5 17. b3 {With a very positional game revolving around the control of c5.}) 4... g6 (4... d5 $4 5. exd5 Nxd5 6. Nxe5 Nxc3 7. dxc3 Qxd1+ 8. Kxd1 $16) 5. d4 exd4 6. Nxd4 (6. Nd5 {Another main line.} Bg7 (6... Nxd5 $2 7. exd5 Qe7+ 8. Kf1 Ne5 (8... Qb4 $2 9. Qe2+ Ne7 10. Bg5 Bg7 $2 11. Re1 $18) 9. Qxd4 Bg7 10. Bf4 f6 11. Re1 $16) 7. Bg5 h6 8. Bf6 Bxf6 (8... Kf8 9. Bxg7+ Kxg7 10. Bxc6 Nxc6 11. Nxd4 $16) 9. Nxf6+ Kf8 10. Qd2 $14) 6... Bg7 7. Nxc6 $5 (7. Be3 O-O 8. Qd2 d5 9. O-O-O dxe4 10. Bxc6 Nxc6 11. Nxc6 Qxd2+ 12. Bxd2 bxc6 13. Nxe4 Bf5 14. Rhe1 Rfe8 15. Ng3 Be6 16. b3 $14) 7... Nxc6 8. Be3 O-O 9. Qd2 d6 10. O-O-O $14