I dont think the debate was as much between intuitionism and formalism as it was between intuitionists and people who thought infinite objects are really real. In particular, intuitionists dont object to "A or not A" for finite sets.
In many cases theories become simpler not more complicated with constructive foundations for mathematics, (for instance, synthetic differential geometry is a much more elegant way of doing geometry). Modern categorical foundations for mathematics, using topos theory, has intuitionistic logic as its natural language.
It is exactly constructive foundations of mathematics which is related to foundations of programming languages, and this in many cases is formalized via the Curry-Howard correspondence which takes programs to proofs and vice versa.
In many cases theories become simpler not more complicated with constructive foundations for mathematics, (for instance, synthetic differential geometry is a much more elegant way of doing geometry). Modern categorical foundations for mathematics, using topos theory, has intuitionistic logic as its natural language.
It is exactly constructive foundations of mathematics which is related to foundations of programming languages, and this in many cases is formalized via the Curry-Howard correspondence which takes programs to proofs and vice versa.