My thoughts are. When you move you d and f pawns in the opening whilst your King is in the centre you create weaknesses on the diagonals to e1. The natural loss of the c pawn to maintain your central pairing means your King can be attacked on both sides before casteling. Once the king is castled away and the initial jostle with the diagonal moving pieces is sorted then moving the f pawn is more advantageous.
Why is La Bourdonnais French so uncommon? (1.e4 e6 2.f4)
White invests a tempo to get a better support of the e5 pawn. But in return, black will get a lead in development, a knight on f5 and quicker pressure on d4. If white can consolidate and develop the rest of the kingside while keeping d4 defended, no doubt white will have a nice game. But these concessions mean that black will get the initiative in return.

Knight is misplaced.
White is put on the defensive.
Black easily develops his pieces.
The advance variation gives the same structure without conceding an advantage to black. Why would masters play the la bourdannis? There's simply no good reason to.
I would suggest you look at the game McConnell-Morphy (New Orleans, 1850). it is the only game known in which Morphy played the French and while it's not a la Bourdonnais white plays f4 on the 5th move while he hasn't developed a single piece

First, you could discard the dogmatic comments about White being worse. He is not: He has grabbed some space, and secured the e5 point, but at some cost, namely self- restriction of the c1 bishop, and difficulties creating the traditional kingside play because of the presence of a pawn at f4.
I guess that Black is fine at the position in #1 as long as he does not castle kingside too early but the whole variation is quite playable- if you want to play it as white, then do it.
An engine may even tell you that white is better: these creepy code compiles just love having more space. But the truth is that, while objectively White is OK, it's more difficult for him to avoid a bad move than his opponent. He lacks a clear plan, and I would be slightly worried about this (although this means nothing at all at the low-to-mid rating scales).
#1
Pawn moves are not developing moves.
In your example of 9 moves you play 6 white pawn moves.
White can afford that, but it makes it easier for black.

#1
Pawn moves are not developing moves.
In your example of 9 moves you play 6 white pawn moves.
White can afford that, but it makes it easier for black.
With the center closed, fast development isn't a big priority.
It might be worthwhile comparing this to an English Opening: 1.c4 e5 2.Nc3 Nc6 (or 2...d6 3.Nf3 f5) 3.Nf3 f5 4.d4 e4 5.Ng5. Later white retreats Ng5-h3 and plays e2-e3. In the French Defense version, black is -1 tempo for being black, and +2 tempi for playing ...Ng8-h6 directly instead of ...Ng8-f6-g4-h6. So by playing 1.e4 e6 2.f4, you are basically letting black be white in an English Opening _and_ have an extra tempo on top. It's certainly "playable". I think Alexander Bangiev even recommended this system against the French in an old repertoire. But most white players would rather do a little more with the first move initiative. Even the much-maligned Exchange Variation is better than offering to take the black pieces with a tempo minus.

It might be worthwhile comparing this to an English Opening: 1.c4 e5 2.Nc3 Nc6 (or 2...d6 3.Nf3 f5) 3.Nf3 f5 4.d4 e4 5.Ng5. Later white retreats Ng5-h3 and plays e2-e3. In the French Defense version, black is -1 tempo for being black, and +2 tempi for playing ...Ng8-h6 directly instead of ...Ng8-f6-g4-h6. So by playing 1.e4 e6 2.f4, you are basically letting black be white in an English Opening _and_ have an extra tempo on top. It's certainly "playable". I think Alexander Bangiev even recommended this system against the French in an old repertoire. But most white players would rather do a little more with the first move initiative. Even the much-maligned Exchange Variation is better than offering to take the black pieces with a tempo minus.
Black's extra move in this structure is a knight at c6, and it's easy to see that the square is far from optimal.
In some lines it might be even worse- e.g. notice the stupid gambit 1.Nf3 d5 2.e4? dxe4 3.Ng5 Bf5. This is a reversed Budapest, with a tempo more for white. However, the tempo has to do with Black's c-pawn (it is still on c7 instead of c5) and this greatly favors Black, actually he is almost winning after just 3 moves.
You can find even weirder examples- like 1.Nf3 f5 2.d3 Nc6!? 3.d4! which looks stupid, but actually is quite annoying.
There's not really a whole lot of stuff on the La Bourdonnais variation of the French that I could find online, mainly because it's uncommon at most levels, and yet I don't understand why it's so underplayed. It seems to me at least to be quite playable for white. I would encounter this from black's perspective in a couple games here and there and think that 2.f4 was inaccurate because of its rarity. But I wasn't ever able to take advantage of it, which led me to dig a little deeper into this variation.
My understanding so far is that the bourdonnais delays d2-d4 in favor of early kingside expansion which is in line with white's plans in the French. Since white is always a step or two ahead of black in controlling the d4 square there doesn't seem to be too much of a drawback. Here's the first 9 top moves of the bourdonnais from the lichess masters' database:
They're very similar positions except for the fact that the bourdonnais has a better defended e-pawn at the expense of an undeveloped LSB. So why is one played far more than the other? Is the pawn on f4 really all that helpful to white in the given context? And if I have a misunderstanding about the goals of both sides in the given position then any help there is welcome too