There have been some recent discussions about a very similar subject in the chat room, but I thought I'd ask here. Are there any businesses that as a photographer you sign on with them, they do the marketing entirely for you, they find clients for you, they take a cut of your profit, and let you be on your way? Just curious if such a group exists.
There sure are. There are literally hundreds of companies which perform these sorts of services, they're called 'Photographer's Agents' or 'Photo Reps' and they're analogous to agents and agencies that actors, directors, models, etc. would hire to manage their business for them.... In fact, a great many agencies 'overlap' between photography and filmmaking, so it's not unusual to see talent rosters that include both photographers and directors (and this is often how photographers such as Michael Bay and David LaChapelle make the transition from photography into the world of filmmaking)...
The one thing that these agencies won't do is your last point: 'and let you be on your way'... I've never heard of a situation (and doubt such a situation exists) where a photographer isn't exclusively tied to an agent or agency via a multi-year contract.
It's not necessarily a 'comprehensive' list (it's quite a dynamic industry and agents/agencies are always going under, merging, etc.), but this list will give you a jumping-off point on the range of agents and agencies that are out there, as well as the level and quantity of talent that they represent.