Well, yes. They're the ones who actually employ the hundred lawyers and PR staff. But that's what it takes, and it's not about clicking some buttons on a Web 2.0 Day Trading UI.
It's really not far from it. You don't need to do a private off-exchange sale (though you can), they do incremental execution (where you break the trade in small batches)
Isn't that literally what institutional trading desks are for? One of the reason to use one is that they can execute a large order without impacting the market price