The mop works fine for me. You might be looking at the uncleanable dirt and grime that is part of the base texture and not a separate overlay. (Hoping that changes in the full build)
Yeah, since I made this comment I realised how useless the mop is now. Not only does it take 6 passes to clean a dirty spot (instead of the 5 previously) - but the sponge cleans everything much more efficiently now when you grab and drag it, compared to how it was prior to 0.7.0.