How frequently do non-programmers break into the software development field, and how do they get in?

Never. The question makes no sense.  People who develop software are programmers or software engineers.  This is sort of asking like "how do people who have never even drawn a stick man get a job as a commercial artist."  The answer is, they don't.  They get the skills either by going to  a school or self-educating.  In either way they need the skills before they can get a job. Are you asking what other jobs there are in the industry?  The answer is a software company has all the same sorts of business people any other company has.  Game development is somehwat special in that there is also a heavy art and design component that people come into from the art side.

Never, in my experience. The handy thing with software development is that it's a highly technical discipline, so it's easy to check for the specific knowledge and skills that go along with it. Things like fizzbuzz programming tests or simple technical discussions will quickly reveal if someone is a software developer or not. Evaluating whether they're a suitable one for a given role is of course a different story and much harder to get right, but by the nature of what we do, finding out if someone is a layperson masquerading as a developer is pretty straightforward.

The simplest way would be to day dream about it. I highly doubt that this happens frequently as people across the world have better things to day dream about. I suggest you hike to the nearest sunny beach or a shady tree in a nice park, take a couple of sheets and pack yourself a sandwich, lay down and try to sleep while picturing yourself working as a programmer without knowing programming. While you are at it, you can also day dream about getting into Manchester United as a left winger while being a person who has never even tickled a football with his right toe.  Yes that should do it. Or ask a better question.