My biggest thing, re: statted systems vs. consent games (because consent games are not systems, so the term does not apply) is where the power lies.
On the latter, the power lies in the hands of Staff and whoever has the loudest voice. Your meek players and those who are not comfortable approaching authority are at a loss. How many people have just quietly left games because they don't feel comfortable approaching Staff?
I mean, shit, I am the person other people come to on games because I'm that person who will just bullshit with Staff on channels and even I have just left games rather than deal with some shit because I've encountered Staff I can't find that level of rapport with. I know people who can never, ever find that level of comfort. Faraday, for example, is one of the most approachable Staffers you will ever meet and I have had to shove people at her and tell them it is totally okay to talk to her and that is on mundane, happy issues and not 'hey this person is being a problem.'
In statted systems, you put power in the hands of your players. You give even your meek, anxious, introverted players something they can hold onto. You give people a voice. You give them something concrete they can hold onto, they can point to, they can back themselves up with. You give your Staff a base to work off of in arguments rather than he-said-she-said and hoping a smart third party or a decent log is available.
I love consent for storytelling and I still engage in a lot of it, even in statted systems, but even then only for people I trust.