No, that's another sort of misconception, also expressed in another comment by WalterBright, which conveniently ignores the reality of most jobs.
It glosses over the fact that employers exercise control over the social relations required for production (of anything larger that can be built by a self-employed person). This happens by virtue of owning all the crucial means of production involved. And that point, where you need to coordinate work of several people, it ceases to be a system of contractors who freely determine their working conditions, and becomes a collective that has a common goal.
So no, it's not case in the U.S., in no economy of the world is majority of production organized into everyone being a little independent contractor who brings (or rents) their own equipment. That would be horribly inefficient (not to mention that people don't want it either, by and large).
There is a clear rebut to this, how can employer own the social relations (required for production), like managerial relationships, when they ostensibly only own the factory equipment? Well, it's like when you own an appartment, you technically only own the four walls, but practically you also enjoy the privacy that comes with it. In a similar way, capitalists owning a factory don't just rent equipment to a bunch workers, but can dictate the whole social superstructure of production, including the redistribution of earnings.