...and is not circular. It makes trees and checks how well they fit the weights they themselves imply. All other comments stand.

Reweighting = iterative/successive/a-posteriori weighting does work as described here, and is indeed circular.

Goloboff PA. 1997. Self-weighted optimization: tree searches and character state reconstructions under implied transformation costs. Cladistics 13(3): 225–245. https://doi.org/10.1111/j.1096-0031.1997.tb00317.x