And in fact mathematics is a great place to dive in, because one very important question -- namely, whether mathematical realism is true¹ -- is really a metaphysical question, and one which can never be answered by empirical observations about the world.
I'd also caution against too much scorn on philosophers going "beyond the reaches of their knowledge base"; historically, philosophy's successes have spun off into entire fields whose origins we now like to forget or at least overlook.
(and, well, I've had a successful career in programming, and also hold a degree in philosophy, and I think the basic observation that lots of people do metaphysics who we don't normally think of as doing metaphysics is a sound one, regardless of how you feel about the paper's specific examples)
¹"Mathematical realism" asserts that truths of mathematics are necessary and discovered, rather than contingent and created, and thus have some sort of existence independently of human minds and human culture. The exact sense in which these mathematical truth-entities would exist is, of course, a metaphysical question, as is the question of what mathematics really is, if mathematical realism is false.
The important distinction between math and metaphysics is that the substantive content of the field of mathematics does not turn on whether mathematical realism is true. Mathematics tells us something about the world by telling us what can be true about the world. We know that there are prime numbers because we know that no such world exists where prime numbers are finite. But this is just saying that a system with certain axioms have certain relationships, or necessarily exclude certain relationships (e.g. whatever axioms of arithmetic you take).
On the other hand, there is no substantive content in metaphysics (at least as far as ontology goes) if the statements do not pick out real entities.
You've just committed the fallacy of begging the question -- you assumed the truth of mathematical realism in this line, and arguably in a couple others.
If mathematical realism is false, various parts of your argument start to crumble.