I don't have a position to defend in this debate, and I certainly don't have any answers. I do have some questions/concerns about some of the definitions, how some of the data is interpreted, and how some realities are perhaps ignored (or abstracted away) during the analysis ... and that's without getting into the questions of what constitutes a STEM degree or a "STEM job". In short, I'm happy to muddy the waters further. Here, in no particular order, are some thoughts:
Not all STEM graduates get "STEM jobs" ... nor need they.
Do the debaters consider jobs in banking and finance to be "STEM jobs"? My guess is that the answer in most cases is no; and yet the banking and financial services industries employ significant numbers of mathematics and statistics graduates. Actuarial positions may be classified as "STEM jobs", but what about people who work designing portfolios or analyzing market trends? Wall Street is a leading employer of Ph. D. graduates in particle physics (see, for instance, this New York Times article), largely because the physicists (claim to) understand the Ito calculus, used to describe physical processes such as Brownian motion but also used in derivative pricing.
My personal opinion, formed well before the 2008 market crash, can be summed up as follows: Handing my retirement portfolio to people who think they can measure particles (tachyons) that exist only at speeds greater than the speed of light ... what could possibly go wrong with that? I cringed when I learned that my alma mater, Princeton University, has a department titled Operations Research and Financial Engineering -- and trust me, it's not the OR part at which I cringe. Personal prejudices aside, though, it seems reasonable to me that a portion of STEM graduates will legitimately be desired for (and hired into) jobs that are probably not considered "STEM jobs", siphoning a portion of our university output away from the jobs specifically targeting holders of STEM degrees.
People have changes of heart.
Years ago, I had a student in an MBA course (now a lifelong friend) who had a bachelors degree in a scientific field (chemistry or something related). She had worked for the Michigan Department of Natural Resources as a lab chemist before deciding that it was not her passion, and that she would rather work in the business sector. Years later, I had another student with a science degree (chemistry? biochemistry? chemical engineering?) who had worked in the pharmaceutical industry before joining the MBA program. After graduating, he worked in management jobs that leveraged his scientific background but would not be classified as STEM jobs. Another MBA student had a degree in nuclear engineering and had served as a propulsion engineer aboard a US Navy submarine.
In fact, a fairly sizable portion of our MBA program consisted of people with undergraduate STEM degrees (and, often related work experience) who had decided to go over to the Dark Side and join the ranks of management. In comparing supply of and demand for STEM degrees, we need to allow for the fact that some STEM degree holders will naturally choose, for reasons other than a lack of job opportunities, to pursue non-STEM careers.
There is more to employability than having the right degree.
An administrator once made the mistake of inviting me to participate in an orientation session for new MBA students, designed to foster camaraderie and esprit de corps. During the session, I remember saying the following: "Look at the person on your left. Now look at the person on your right. Statistically, one in three people is an asshole ... so if it's not one of them, it's you." (I made the statistic up, based on my experience growing up in New York.) The point I was making then was that candidates would be required to work in teams, just as in industry, and it was naive to assume that it would always be easy to get along with all your teammates.
Sadly, though, it is also true that some people just cannot coexist at all with other workers. Some are chronically absent or late. Some need to be nagged, wheedled or hand-held just to get things done. Some are larcenous. I see no inherent reason why a STEM degree would preclude any of those possibilities (and in fact I've met walking, talking counterexamples to that conjecture). Those people will often "fail" a job interview, no matter how solid their technical credentials, or they will be involuntarily separated from an employer in a way that follows them through subsequent interviews. Thus we have to allow for some slice, hopefully small, of the STEM degree holders to be unsuitable for employment in STEM jobs (unless the recruiters are desperate).
Educational standards in the US ain't what they used to be.
During my graduate studies in mathematics at Michigan State University, I was a teaching assistant. Like most TAs, I began teaching recitation sections of a large service course, whose title was (approximately) "Introduction to College Algebra". The course was taught in a large lecture by a faculty member. The primary roles of the TAs handling the recitation sections were to answer questions and grade homework and exams.
A year or so after I arrived, we were joined by a new graduate student with a bachelor's degree in mathematics from a "prestigious" university. (I shall not name the university, so as to avoid embarrassing the fine folks in Ann Arbor.) He too had a teaching assistantship. Fall quarter of his first year, he was assigned to teach a couple of sections of the introductory algebra course. Winter quarter he was pulled out of the classroom, and his sole duty as TA was to work out the answers to the problems in the textbook ... because the Powers That Be had discovered that he could not answer simple algebra questions in class (and not, apparently, due to stage fright). Had he chosen to work in industry, rather than going straight to graduate school, and had the recruiters done their jobs well, he might have contributed to the statistics representing STEM degree holders not working in STEM jobs.
Employers frequently complain (particularly when lobbying for more H1B visas) that they cannot find a sufficient number of STEM degree holders with the "right skill set". We can argue about who bears responsibility for a genuine lack of the right skills: universities with outmoded curricula; employers unwilling to pay for training; or degree holders unwilling to upgrade their skills on their own time. We can also speculate (and many people do -- see the comments on the IEEE Spectrum post) on how often the "right skill set" translates to "willing to work cheap". We also need to accept that in some cases the "right skill set" translates to "actually knows the subject matter for their awarded degree", and that this is not a given.