It does follow. If you're hiring people who are equally attractive to you, and one is attractive in part because of the degree, then for them to be equally attractive they must be worse on your other desired qualifications - such as demonstrated competence. My claim is that demonstrated competence is more important than the degree. Therefore of those two candidates, the one without the degree usually turns out to be better.
This is actually true about any discriminated against group (which people without degrees are). On average the discriminated against group may be worse (for whatever historical reasons), but the ones who are good enough to become seriously considered despite that are actually better than the ones you would consider equally desirable. Therefore if you're on the fence about a decision, you should prefer the one who lacks the most obvious signals like degrees, etc.
This may sound like an abstract and weird hypothesis. But it is a testable one. For example see http://hbswk.hbs.edu/item/6498.html for evidence that being willing to hire women into management in a culture where women are discriminated against results in better financial returns for the company that is willing to do so.
As for your hypothesis that people with a degree are more likely to have the required skills, that depends heavily on the job. Certainly if you're going to hire someone to work on a compiler, you'd prefer people who know how compilers work. Which gives a big edge to CS grads. But for general software development I'd prefer someone who can quote Code Complete back at me than an average CS grad.
(Disclaimer, I have a masters in math, and almost finished my PhD. If you are looking to hire me, and you have an alternate candidate without my academic qualifications who you think is a tossup compared to me, then yes, I am saying that you should hire them instead.)