I've been experimenting with this method for a little bit and thought I'd mention it in here, it seems pretty interesting. The steps are:
1. EOline
2. Left block
2b. CP (optional)
4. BR block (cross piece + pair)
5. U-layer block (2x2x1) and parity fix
6. Permutation of last 3 edges and last 4 corners
7. Corner orientation
So, assuming you don't do CP, you get to F2L-1 through blockbuilding. Then you can use the open slot to make the U-layer block intuitively, this isn't that hard. Once this is done, place the block and check the last three edges for parity. If there is, the solution is intuitive (the U center must be rotated 90 degrees), but you can use an alg for this. I use U R U R' U' R U' R' U2 R U2 R' U'.
After parity, solve EP (using U' R U R' or its inverse) and corners can be placed using the same trigger in sets of three. For example, doing the trigger three times swaps UFL and DBR, and swaps UFR and UBR. Rotating and doing the trigger left-handed produces a similar result, so corner placement isn't bad, although if CP was done earlier, the triple trigger is only needed once if at all, and without rotations.
CO to finish, I just use a CLS alg for this which only affects two corners.
Example solve: B2 L' B2 U2 R D2 B2 R D2 R' B2 F' R' D U2 F2 R2 D' B F' U
EOline: F D U L B' R' D2
Left block: U2 L' U L2 U' L' R U2 R U L' U' L
CP: L' U R2 U' L
BR Block: U R U R2 U' R U R2 U' R2
U Block: R U R' U2 R U2 R' U2 R U' R' U' R U2 R'
Parity fix: (U' R U R') U R U R' U' R U' R' U2 R U2 R' U'
Corner Placement: (U' R U R')3
CO: U R U' R2 U' R U' R' U2 R' U2 R' U' R U' R2 U2
I think this method really shines in its stats, the chances of parity skip is 1/2, the chances of edges being cycled is 1/3, and the chances of corners being placed (assuming CP is done beforehand) is 1/2, so the total probability of only having to twist corners after blockbuilding is 1/12. I'm not sure about the probability if CP isn't done, but this by itself is really good.