How a matchmaking algorithm saved lives

You are right that the general problem is intractable; a lot of the advances in these match making algorithms come from better heuristics and computing power.

Not sure this particular algorithm has anything to do about economics, except its author is an economist.

macroeconomics is all handwaving

That is actually not related, but the beginning of an entirely different field of mathematics. Design theory is the study of mathematical objects called combinatorial designs[1], which are basically fancy ways of partitioning finite sets into subsets with a pre-specified structure. The thing you linked to is a specific instance that shows the existence of designs in a few specific cases.

While Kirkman essentially posed the first design theory question in the 1850's, both the authors of one of the standard textbooks[2] in the field are still alive and (IIRC) still working today!

[1] https://en.wikipedia.org/wiki/Combinatorial_design . Incidentally, the field is called "design theory," but it's not a great search term outside of MathSciNet.

[2] http://www.amazon.com/Design-Edition-Discrete-Mathematics-Ap...

Not that I'm aware of. (I've read a fair bit on the topic.)

The basic Gale-Shapley algorithm was published in 1962.

https://en.wikipedia.org/wiki/Stable_marriage_problem