It appears to be 4...Bb4+ that's the unusual move here. The chess.com database only shows one game with it, and that's the game that continued with 5.c3. When you have the position after Bb4+ on the board and it shows 5.Nc3 with a bunch of games, it's actually showing games that transposed into that position - the Nc3 was played earlier in the game.
You're playing a very sharp opening line that only works thanks to the option of playing 10...g5!. You should only enter this type of position if you're aware of what you're doing. Otherwise you'd be better sticking to quieter, easier positions. If you choose to play a theory-heavy line, then you should definitely know the theory first!
I am pretty sure the first 10 moves are pretty ok, but after Kg2 you are supposed to play either 0-0 or g5 (maybe some other moves are fine too; fxe fxe 0-0 is also not too bad I suppose), but Kf7 looks weird. Generally, in this line black is indeed suffocating unless he sacs a knight on e5 to open up files in the middle and then attack the white king. I ...
As suggested, I'll convert my comment to an answer. Given this format however, I reckon a list makes more sense. So let's see how the move 5.f4 changes the position:
5.f4 is a slow move that wastes time for white in a line where White already invested a tempo on 3.e5.
5.f4 makes it easier for Black to develop their knight on g8 via h6, from where it lands ...
It is certainly playable. The main drawback is that you give White the choice between two very different mainline openings:
3.Nf3 transposes to a Sicilian, where Black is already committed to e6 (Kan, Taimanov or Scheveningen, but no Najdorf, Sveshnikov, etc.).
3.d5 reaches a Benoni structure and may transpose to a mainline Benoni if White later plays c4. ...
Another idea of the two knights variation is to provoke 3...d4 and get some kind of reversed Old Indian. Some players like these pawn structures, and it also may not be blacks taste to be the one with the space advantage.
1.e4 e6 2.Nf3 d5 3.Nc3 d4
( 3...Nf6 )
Not everyone regards "Bad" Bishops as unmitigated evil. Read "The Secret Life of Bad Bishops" by Eben Lund to discover the subtlety of the issues.
If you will play the French you must do this. Hopefully you will understand them better than your opponent.
d5 is a very good Benoni for white (if there is such thing as a not-very-good Benoni for white) exactly as Arne mentioned, because white can maneuver his knight to c4. Usually arises from a different move order though, 1. d4 c5 2. d5 e6 3. e4, and this is the reason why 1. ... c5 is not great against 1. d4. You'd want to wait for that pawn to arrive on c4.
You've pretty much answered your own question. Black can go 3...Nf6 and go for the mainline, but White is managing to avoid some other lines like the Winaver.
Anyway your question could be reversed just as well: why play the main move order whne you could go for the Two Knights French all the time instead?
Look at the stockfish lines after Nge7, or, Bd7, or a6. In all of them, black play Na5-b3 sooner or later, and for that the knight on b3 needs the queen's support. In that regard, Bb4+ is really not helping after, say, white plays Nc3 and the bishop on b4 blocks the queen from controlling the b3 square. Apparently Stockfish likes the Na5-b3 maneuver so much ...
I have trouble telling what's theory and what's my practical notes, but one thing immediately comes to mind.
Castling queenside can be nice in the Winawer, but it may be almost necessary in the MacCutcheon variation:
1. e4 e6 2. d4 d5 3. Nc3 Nf6 4. Bg5 Bb4 5. e5 h6 6. Bd2 Bxc3 7. ...
well, having seen f5 in my game, I would probably reply with h4 and then bring a knight from b1 to f4 and a rook to g3 (if necessary). I remember something like this being discussed in one of the Dvoretsky's books as a response to early f5. I don't think black has any potential of developing a king side play after that.
Of course, in short time control ...
I think the most "French-like" reply to 1.d4 is the King's Indian Defense. By playing an early ...e6 and ..d5 you're making the same moves as in the French, but you hardly ever get a similar type of pawn structure.
However, with the King's Indian, you'll often get a closed center with a chain of blocked pawns on it, with each side playing on an ...
Depending on what database you look at, all three moves (5.c3, 5.Nc3, 5.Bd2) are playing in this position. Nc3 and c3 are the most common, but Bd2 is played sometimes.
A lot of the play in the French Exchange is centered around the central squares and the open e-file. Quite often, White will play c4 and Nc3 and try to deploy their bishops at d3 and f5 or g4.
I would say that 5.Nc3 is better than 5.Bd2 because 5.Bd2 would allow Black to exchange Bishops very early in the game. It is a general principle that more piece exchanges lead to less complicated middlegames and to more chances of draw, which is naturally something that favors Black. The move 5.c3 does not look optimal to me since, as you said, it blocks ...