I've got an idiosyncratic addition to these answers:
(a) Regarding contemporary mathematical uses of the Cardano formula, I coauthored an article that "uses" it! The role it plays in the paper is essentially as inspiration: we noticed that the set of roots of a family of polynomials we cared about had the same form as the roots of a "depressed" cubic as given by the Cardano formula, and our search for why that was happening led us to a deeper understanding of our family of polynomials. (They turned out to be generalized Lucas polynomials!) As an addendum, we note that the same Cardano(/del Ferro/Tartaglia) solution method for the cubic can also be used to solve the polynomials in this family.
Aside: the article has been accepted for publication in the American Mathematical Monthly and will come out sometime next year. The review process led to a substantial revision that I believe vastly improves the article, but I can't post it because Taylor & Francis owns it, so we have to make do with the arXiv preprint till publication. That said, the treatment and use of the Cardano formula aren't really changed.
(b) Regarding historians of mathematics, do you know Michael Barany? He is a historian of mathematics active (and generally responsive) on Twitter, so he may be able to point you toward people with the relevant expertise.
(c) I am not a historian of math, but I have read Cardano's Ars Magna (in English translation), so if you need a fact-check on something that's in there, I might know the answer. Email me? Otoh, it was a library book and it was a decade ago, so maybe this is a dumb idea. And I'm unlikely to know anything about the del Ferro/Tartaglia/Cardano/Ferrari(/Bombelli??) saga that you don't already know.
(d) Addendum: this is of possible relevance, although it is not an answer to the question and you probably already know it. Picking up on Dave Renfro's comments on the OP, my understanding is that one of the Cardano formula's important historical roles was in getting mathematicians to take the complex numbers seriously. I learned about this from the first chapter in Tristan Needham's book Visual Complex Analysis, and Needham also points to John Stillwell's book Mathematics and Its History. Imaginaries are first discussed in print in Cardano's book but in a context unrelated to the cubics, and the discussion is highly speculative and Cardano is really unsure there is any point to what he is saying. (See the link above on Ars Magna which is a blog post I wrote that discusses this toward the end.) However, 30 years later, Rafael Bombelli discovered that certain real cubics with real roots, for which the Cardano(/del Ferro/Tartaglia) formula appears to "break" by requiring the extraction of a square root of a negative, can still be solved by the formula by going ahead and just treating the extraction of the root of a negative as something you're allowed to do: all the weird imaginary numbers you get that way cancel out of the final answer, and the final answer is the correct real root. According to Needham, this was the real impetus for mathematicians starting to take complex numbers seriously.
(I guess this is just registering a perhaps-lesser-known contribution that the cubic formula made to math history. I suppose while I'm brainstorming things of this type that you already know, I should mention the better-known fact that it [and Ferrari's quartic formula, also published in Ars Magna] are what set off the search for the solution to the quintic, which is what led to the development of Galois theory, therefore group theory, therefore abstract algebra as a whole, so between complex numbers and abstract algebra, I guess pretty much all of modern math is downstream from the Cardano formula somehow!)