I think you have made your point, but in kind of an around the corner way.None of the Forsaken or Shadar can be proved to have visited Far Madding,They send others there. Slayer and Fain can function there but neither of them needs the power

 

Haran is able to to totally shield the Chosen in much the same way that FM does the one power. It just isn't there anymore Interesting thought !

 

Now as to someone stealing the CK sometimes we forget that Rand is the strongest Tavern in memory and that doesn't have anything to do with shutting down what the Dragon can do.

 

As to why he went to FM These guys can Travel. he needed them off his back so he could clean up the male side of the power. Remember they followed him to FM It was his trap! He just chose a place where he felt he had the best odds to take them out one at a time. Then being Tavern got him out of Fm with the Protection he needed while doing the job

Well Shaidar Haren may not have been able too... we know that normal Myrrdraal avoid stedding at all costs, it may have something to do with their nature as constructs, similar to the gateway effect.