And unless you have been hired by a company before, almost every application for a consulting gig starts with sending a resume, which is used as a first filter for the company potentially hiring you. Now the difference is that for many consulting gigs, you have to do a very specific thing - so they might weight experience more than degree. But that's a chicken-and-egg problem, to be able to get (more) experience, one must have experience.
I agree, indeed it is a chicken-and-egg problem, but I don't see that much as a big problem. From 1994 to 2000 I've contracted 60 people and to be able to hire the better ones, I had to read and analyze hundreds of resumes. I discussed this work with several other (personnel) managers and almost everybody does stress the importance of experience, but there are several forms of experience.
The form most considered is experience in a paid job, for an employer. Quite often, applicants for a job at my cmpany, didn't have such experience, but they had exerience in working at home or (as a student) at university or high school. And they could show that experience: websites, database-driven, several scripts, server logs and statistics.
For me, and quite a lot of managers of small companies, a degree is absolutely not important, and neither is experience in a paid job. Just be able to show you have experience.