They are not insightful in new ways about human cognition at all.
I’m not sure how you reached this conclusion but it’s completely wrong. The opposite is true: algorithms offer the deepest insight into human cognition ever. This is formalised in the Church–Turing thesis. The conjecture is about the computability of functions but it has a very important corollary about our intuition of computability.
It effectively states that the human mind is in principle limitless (sometimes termed a “universal constructor”; not to be confused with the similar but distinct concepts of “universal assembler” and “von Neumann universal constructor”).
If the hypothesis is true, then (according to David Deutsch, The Beginning of Infinity) the (human) mind is unique in all of the known Universe: it alone is capable of causing a sustained effect over the rest of the physical Universe.
The argument becomes a bit convoluted but it essentially reduces to the point that everything else on our world, our solar system and even our galaxy is limited in scope: go away far enough and its effect vanishes. A creative mind (like the human mind) alone is able of extending its own physically imposed limits, because our minds are able of generating knowledge.
If you’re interested in human cognition, I recommend you read Deutsch’s book which explains this in great detail. And all of this follows directly from … the insightfulness of algorithms, in particular the Church–Turing thesis.
Apart from that, algorithms are interesting once they cause you a shift in cognition. For instance, once you first understand recursion, or dynamic programming, or transitivity, it’s as if a small light goes on in your head – you have an insight. So algorithms are, in the primary sense of the world, insightful.