Perhaps, but the recruiter's cut is based on your salary, and they also usually know how much the client is willing to pay - and quite possibly what they're previously paid for people of your calibre.
If I ask for more money and the company says no, I can walk away and keep my current job, losing nothing. If a recruiter walks away they lose their huge commission.
Companies know this, so recruiters have no negotiating leverage.
Companies know this, so recruiters have no negotiating leverage.