I think it's a great idea for companies to have a contract specialist on staff. Contracts are quite complicated, and naturally each party to the contract is going to be looking out for their own interests. It never hurts to have an expert in your corner to make sure your company's interests are adequately represented.
I wonder if there are people who do contract specialist work as freelancers? I bet there are plenty of small businesses that might hire someone like this occasionally. Or maybe those businesses just have their lawyers look over their contracts instead?