Bird c4 variation?

First of all black is not forced to capture on c4, so be careful saying the main idea is that black cooperates.
One way to play a dutch / bird is to play for e4. You don't move the e pawn in the beginning, you play d3, finachetto kingside, then try for e4 in 1 move. In that case if you play c4, not only does it spend a move that has nothing to do with e4, it could create holes on d4, e3, or even e4 if black captures on c4 and you recapture with the d pawn.
In setups where you're not going for e4, it's similarly a little slow and you lose control of d4. For example
And in the final position notice how if white moves the d pawn, it weakens e3. If he moves it to d4 then it blocks the bishop and weakens both e3 and e4. Probably white will try to leave the pawn on d2 for a while.

First of all black is not forced to capture on c4, so be careful saying the main idea is that black cooperates.
One way to play a dutch / bird is to play for e4. You don't move the e pawn in the beginning, you play d3, finachetto kingside, then try for e4 in 1 move. In that case if you play c4, not only does it spend a move that has nothing to do with e4, it could create holes on d4, e3, or even e4 if black captures on c4 and you recapture with the d pawn.
In setups where you're not going for e4, it's similarly a little slow and you lose control of d4. For example
And in the final position notice how if white moves the d pawn, it weakens e3. If he moves it to d4 then it blocks the bishop and weakens both e3 and e4. Probably white will try to leave the pawn on d4 for a while.
Those are good points. First, I personally think that black not capturing on c4 is the way to go, and because of that the game i posted isn't a great example. Secondly, I was only referring to this c4 push if black has played Nc6 or isn't able to play c5 and create the c/d pawn duo in the 5th rank
I've been studying the bird for a little while now, and there seems to be only a couple variations that actually have theory. The polar bear system, the Dutch variation, and the stonewall. There's also froms gambit. However, I can't find any theory on the variation below. The main idea is to leave the d pawn on d2, and push with an early c4. It gains space on the queenside, opens up the c file for the a1 rook, and greatly improves whites light squared bishop. It can be used against multiple set ups from black. Is there a reason why there's no theory or master games with it, but there are for the other variations of the bird?
here's a low level example:
1. f4 d5 2. e3 Nf6 3. Nf3 Nc6 4. b3 e6 5. Bb2 Bd6 6. c4 dxc4 7. Bxc4 Nb4 8. a3 b5 9. Bxb5+ c6 10. Bc4 Ne4 11. axb4 Bxb4 12. Bxg7 Rg8 13. Bc3 c5 14. O-O Bxc3 15. Nxc3 Nxd2 16. Nxd2 Bb7 17. e4 e5 18. fxe5 Qg5 19. Bxf7+ Ke7 20. Bxg8 Rxg8 21. g3 Bc8 22. Rxa7+ Bd7 23. Nd5+ Ke6 24. Rf6+ Kxe5 25. Nf3+ Kxe4 26. Nxg5+ Rxg5 27. Nc3+ Ke5 28. Qd6#
(Can someone actually insert that game? I can't from my phone)