Why is intuitionistic reasoning blessed as the "universal reasoning," when there are other ways of doing logic that permit even fewer inference rules?
It is just a fact of life, without any room for philosophical preferences, that the largest common denominator of all toposes is exactly intuitionistic reasoning, not more, not less.
But there are, besides toposes, also other kinds of mathematical structures which can be regarded as mathematical universes! And the largest common denominator of those other kinds can be more or less than intuitionistic reasoning.
Going up, we have for instance models of ZFC. By definition, their largest common denominator is ZFC, so more than intuitionistic reasoning.
Going down, we have the so called "arithmetic universes". Their largest common denominator is "arithmetic type theory", a predicative flavor of intuitionistic reasoning. (To a very rough first approximation which doesn't at all do justice to this intriguing topic, "predicative" means "no powerset axiom". The terminological convention is that by default, "intuitionistic reasoning" refers to impredicative intuitionistic reasoning.)
And then there are a couple other kinds still.
That said, toposes with their impredicative intuitionistic reasoning do occupy a sweet spot. They are sufficiently general to yield useful applications in several branches of mathematics while not being too general.
I think you mean 20th.
There are also even wilder flavors of mathematics which are not set-based. In those we can have various dream results which directly contradict the mathematical canon (but are internally consistent and have a certain precise relation to the ordinary mathematical world). For instance we can have that every function is computable by a Turing machine, that every real function is continuous or that the reals include infinitesimal numbers. An introduction to these flavors, aimed at philosophers of mathematics, can be found here: https://rawgit.com/iblech/internal-methods/master/paper-film...