Not sure if anyone has come up with this idea. Would not be surprised if it has.
It's a small change to CFOP but opens up a ton of different options for F2L. This would not work every solve and so would just be an option to look for if it can be done fast at all.
Solve 3 cross edges with the empty edge in front (otherwise look ahead is very hard)
Solve the back two F2L slots. Because you aren't concerned about the front cross piece, you can solve it similarly to SB in roux. Because the back cross piece is important, there are limitations to using M moves but they can still be useful (see example solve below)
Next, you have a choice. You can,
1) Put in the last cross edge and solve the last two F2L slots normally.
2) Continue to not care about the last cross edge leaving many of the same options available as well as easier insertions (F' instead of F' L F L' etc) and then put in the last cross edge. I usually put in the last cross edge before inserting the last F2L slot and after pairing it up.
NOTE: you will want to try to keep the last cross edge out of it's place. Especially if it is flipped.
Lastly do OLL and PLL.
Here is an example through F2L. I haven't checked to see if there is a good solution normally with this scramble so this may not be the best solution.
I'm not sure if this can be fast, but it's fun to play with.