Has 3.Bc4 in the King's Gambit Accepted really been refuted?


It's suboptimal but not refuted.
It's not even suboptimal but as good as 3. Nf3.

Kasparov was forced to play the Bryan counter-gambit as it was decided by the organisers as part of a set of theme games. After losing in 15 moves he left the stage furious at being forced to play it. However, he co-wrote Batsford Chess Openings 2 with English GM Ray Keene and they gave only a slight plus for white in the Bryan! This was the opening used in the immortal game, Anderssen - Kieseritzky. Probably Kieseritzky had pleasant memories of his win over Schulten.

Named after American player Thomas Jefferson Bryan who frequented the chess cafes of London and Paris in the mid 1800s.

According to Shaw who published extensive analysis, 3. Bc4 is indeed refuted. It offers practical chances but not against a well-prepared, high-level opponent.
Kasparov's 3. Nc3, though not the optimal try, was a clever attempt to avoid the more positional alternatives Black has in response to 3. Nf3. The King on e2 is surprisingly often not misplaced in these positions.
In a long game, Kasparov probably would've played 3. Nf3, that is if he even would've chosen the King's Gambit in the first place. But against that Sergey would've had a gazillion choices.

Thanks for the link, IM pfren!
melvinbluestone: there's also the Cunningham where 1. e4 e5 2. f4 exf4 3. Nf3 Be7 4. Nc3 begs for Black to play 4. ...Bh4+. This used to be my most common way of meeting the Cunningham (and was also a recommendation of Shaw's) back when I regularly employed the KG. It was one of the funnest variations to play in the KG-jungle, IME.
Fair attempt at trolling, I guess. Anyone taken in by that joke article might want to look at its publication date.

I posted a topic on this about a year ago asking for the same information. I only got one useful reply, which was by jatait47:
"Shaw's 1 e4 e5 2 f4 exf4 3 Bc4 Nc6 4 d4 Nf6 5 Nc3 Bb4 6 Ne2 f3! 7 gxf3 d5 8 exd5 Nxd5 9 0-0 Nxc3 10 bxc3 Bd6 line was analysed on ChessPublishing a year or so ago. White was reckoned to be okay after 11. Qd2 and 12. Qg5 (Stefan Bücker). White doesn't have any advantage, but then White doesn't have any advantage in any lines of the King's Gambit, so no worries there"
A good position to analyze if you're serious about 3.Bc4.

That's the point, intermediatedinoz. White cannot claim an advantage in the position. But he's not busted, either.
Wrt your line I don't think you're missing anything, but what's the followup? Black has a nice Knight on f5, which is good for them. But White still has the N and the LSB which can trade themselves for it, so it's not eternal. And while f5 is a nice square, it's not directly attacking anything from there. So White, atleast for the time being, can certainly play around it and do other stuff on the board.
In your line I might continue with 15. Bf4 as White, with the somewhat double-edged idea that if I get to trade the DSBs, ...c5 won't be annoying, and I can start taking open lines for my rooks, without Black being able to immediately exploit weak dark squares on the c5-g1 diagonal.
I suspect White could also consider other 14th moves, such as f4.

11. Ng3 (-0.29) or 11. Qd2 (-0.35).
Stockfish 8 doesn't think it's a refutation. But sure, playing with White's pawn structure would be annoying and difficult for a human.
Capablanca was not a fan of the King's Gambit for white but he recommended Bc4 as best for white. Bc4 is a perfectly good move.

I was curious about Kasparov's surprising choice of the KG in his blitz game with Karjakin in St. Louis. And his 3.Nc3 was just plain weird. I think GM Rensch was being diplomatic when he called it "a rare line", instead of just a plain and simple "inferior line". But he managed a draw with it, so maybe, as Danny said, not a bad choice for a fast time control.
OMG! Kasparov played the Mason Gambit! What a hero!

Suffice to say that Shaw's "refutation" of 3.Bc4 has been "refuted"...
White is not worse in Shaw's line. Here's some information about it:
http://200opengames.blogspot.gr/2017/02/012-bust-to-bishops-gambit.html
That's my blog. Thanks for the link .
Yes, in my opinion White faces more serious challenges in the Bishop's Gambit than 3...Nc6. But then White faces challenges in virtually all lines of the King's Gambit. It's an opening you have to want to play, in full knowledge that White has no theoretical advantage at all and may often come out standing worse, but.... in the kind of random position you like to play, which usually counts for more than a theoretical advantage over the board anyway.

So what's the story: In the KGA, after 3.Bc4, black plays 3...c6 and wins?
LMAO

Kasparov's 3.Nc3 is also known as the Mason-Keres Gambit. Contrary to popular belief, it is not James Mason the British actor the name refers to, but Charles Mason, who, along with Jeremiah Dixon, surveyed the demarcation line that today generally denotes the separation of "north" and "south" in the eastern United States.
No, it is named after James Mason, the Irish chess master of the 19th century, rather than the English actor. Though the only game in the databases where Mason actually played 3 Nc3 was a loss to Rosenthal, Paris 1878. Keres had far more success with it in his 1930s postal games.