Also, if they are not doing it off-chain as you say, how on earth they are able to withhold a transaction I send from my non-cartel wallet.
They can't. He is saying you (and 99% of other Bitcoin users) will be stupid enough to use their fee withholding cartel wallet. Of course the OP isn't stupid enough to do that, just everyone else in the world is. The OP needs to save you from yourself.
I am saying the masses don't know and don't care. They are not as paranoid as we are. They just want to click a button and order their Pizza. They have no clue how it all works behind the scenes. The go to Dominos Pizza and click the button the website. They don't go searching for a Bitcoin client.