MacCutcheon refutation of 4.Bg5?

Is the MacCutcheon a 'refutation' for 4.Bg5 in the Classical French? I have done some engine analysis and it seems that all White can get is draws by perpetual check, and if White doesn't go for those. Stockfish/Komodo 10 consider Blacks position already slightly better by 0.20 to 0.40.

Is this be the reason that 4.Bg5 is no longer seen against the classical French in top level chess?

No

Having a small advantage of 0.2-1.4 as Black doesn't mean that you've refuted the opening White played. Plus, engines are known for not being very useful in the early stages in the opening.

well if an opening leads to even a small edge for Black, White failed in the opening, as White should aim for an advantage.

and I'm referring to lines that are considered 'main lines'

for example:

€: I think the claim 4.Bg5 is refuted by the MacCutcheon can be backed up by the fact that most of the recent french defense repetoire books, made the MacCutcheon on of their main recommendation:

Aagard - playing the french / vitiguov - the french defense reloaded / Antic - The Modern French

I do find that variation the hardest to play against as white out of all the French lines to be fair.

I found myself a way around the MacCutcheon, as I have been playing 4.Bg5 against the Classical French so far and picking up 4.e5 as new move is simply too much to study at the moment.

My approach is simply to play. after 4... Bb4, 5.e5 h6, 6.Be3 Ne4 and now 7.Ne2. not that this promises any advantage, but as I play Ne2 against the Winawar so I can get structures that I am familiar with.

while this is a nice game. Black misplayed the position. a dangerous plan after 11...c4, 12.Be2 is 12... Qb6 with the idea of playing Qb2 and winning the a pawn. and that slow assault on Blacks king is best defended by evacuating the king to the queenside. not keeping it on the kingside.

waiting for the rook to leave the first rank makes sense Nc6 is on the agenda anyways. and no way in hell black will get mated, if he defends accurately or simply leaves the danger zone moving the king to the queen side. if white really just gave away its a-pawn, then blacks a pawn will simply move ahead to promote. with all whites pieces stuck on the kingside shooting blanks thats going to be pretty easy.

Based on engine analysis, I'd go a step further. White has no advantage after 3 Nc3 according to the engine analysis I've seen. The only move that gives white any kind of an edge is 3 Nd2. How seriously one should take the engine analysis in the typically closed positions that arise is another matter, and not one I care to speculate on. But after 3 Nc3 Nf6 it's nothing but triple zeroes as far as Stockfish is concerned. Then after 4 Bg5 it goes for the MacCutcheon.

Even if black could teleport the queen to b2 in one move how is that winning the a pawn?

Here's a "secret" from someone who plays the French Tarrasch with White: Black can equalize in a few different ways after 3...c5.

Interesting game. I was trying to make 45.Qe7 Qc7 46.Bf8 Rxe7 47.fxe7 work because of the threat 48.h7 but he just can play 47. ...Qd7 and you're busted.