Need help with 4x4 rescramble

So I was recently introduced to rescramble by J perm's video on the topic, and I tried it on the 4x4 cube, only to have center parities. I was wondering if there is anyway to get around this problem, because I can't figure out how to.
More needs to be said/written about 4x4 rescramble. In addition to the center parity, there is also a "corner parity" on the last layer. Also, the standard PLL parity alg messes up the cube, as do many/most of the standard PLL algorithms. And don't get me started on the unexpected difficulty of the 5x5 rescramble. (I'm speaking of trying to reverse engineer CFOP--perhaps a blind method is better for rescramble.)
 
So I was recently introduced to rescramble by J perm's video on the topic, and I tried it on the 4x4 cube, only to have center parities. I was wondering if there is anyway to get around this problem, because I can't figure out how to.
If you're okay with messing up the edges, you can use the alg Fw' U' Rw Fw Rw' U' Rw Fw' Rw' U2 Fw U' to swap the top right and bottom right center pieces of the U face (while leaving the other centers alone).

Personally, when doing a 4x4 rescramble, I first follow Yau4 (with the above alg, if needed) to solve centers and edges and cross. Then I solve the first "two" layers as usual, orient the edges of the U layer (using the usual OLL parity alg, if necessary), solve the corners (corner parity may occur!) using beginner's method, then solve the edges (using supercube-safe PLL parity alg and PLL algs).

If anyone is interested, I can post a video and my algs. It takes me about 8 minutes, but I'm definitely not a speedcuber. I think sub-1:30 is attainable by you experts.
 
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