Why isn't 6.Bc4 considered by theory in the 5.Bxf7+ Traxler Counterattack?

Traxler is too rarely played to anybody care. Bd5 and Bb3 are old main lines. Bc4 is actually Lc0 main line on pretty solid depth. One obvious advantage is easier handling of future pin while I don't think the bishop is so exposed there. From b3 it could actually be easier for black to hunt it with Na5. So it's great move in position you will probably not have on a board anytime soon. The old reasoning behind not playing Bc4 makes sense too, of course, but try to explain it to Leela... If it was in more frequent position, it would be played at least as much as the 2.Nf3 3.d3 Caro Kann, very fresh AI invention too.

Comparing 6.Bb3 and 6.Bc4, the drawbacks of the latter are obvious since the bishop is not protected and can become a target to attacks:

• either by a black piece: Ne5, Na5, Qc5, Qd4...

• or, more probably, by the d-pawn: the typical ...d5 will gain a tempo for development

On the other hand, I can see no advantage of having the bishop on c4 rather than b3. You will hardly ever want to retreat to e2, nor to play a queenside fianchetto.

Given how important each tempo can be in such an complicated line, it is understandable than no strong player has ever ventured 6.Bc4. Probably there isn't a direct refutation, but you might regret having your bishop exposed later on.