The (actual) article has a fairly detailed literature review in the introduction, and makes it pretty clear that the main idea was sort-of known already if you squint - but it looks like nobody had put the whole theory together elegantly and advertised it properly. The fact that they couldn't find some natural slices of the hyper-Catalan numbers on OEIS supports that.
The proof they give that the hyper-Catalan series solves the Lagrange inversion problem is very good from a pedagogical point of view - I don't think I'll ever be able to forget it now that I've seen it. The only thing this paper is missing is a direct, self-contained combinatorial proof of the factorial-ratio formula they gave for the hyper-Catalan numbers - digging though the chain of equivalences proved in the references eventually got too annoying for me and I had to sit down and find a proof myself (there is a simple variation of the usual argument for counting Dyck paths [1] that does the trick).
Another thing to note is that the power series solution isn't just "a power series" - it's a hypergeometric series. There are lots of computation techniques that apply to hypergeometric series which don't apply to power series in general (see [2]).
The proof they give that the hyper-Catalan series solves the Lagrange inversion problem is very good from a pedagogical point of view - I don't think I'll ever be able to forget it now that I've seen it. The only thing this paper is missing is a direct, self-contained combinatorial proof of the factorial-ratio formula they gave for the hyper-Catalan numbers - digging though the chain of equivalences proved in the references eventually got too annoying for me and I had to sit down and find a proof myself (there is a simple variation of the usual argument for counting Dyck paths [1] that does the trick).
Another thing to note is that the power series solution isn't just "a power series" - it's a hypergeometric series. There are lots of computation techniques that apply to hypergeometric series which don't apply to power series in general (see [2]).
[1] https://jlmartin.ku.edu/courses/math724-F13/count-dyck.pdf (for instance) [2] https://sites.math.rutgers.edu/~zeilberg/AeqB.pdf