Setting philosophy aside, one of the major applications of constructive logic is that it's the internal language of toposes (and related kinds of categories). This usually simplifies proofs considerably and can produce new results.
Some examples, Joyal and Tierney's An extension of the Galois theory of Grothendieck, CJ Mulvey's papers from the 70s (Intuitionistic algebra and representations of rings for example) and Blechschmidt's A General Nullstellensatz for Generalized Spaces.
Johnstone's Sketches of an Elephant often has both internal and external proofs for comparison and it's easily seen that the internal version is both easier to write and to understand.
For an article explicitly focused on using internal languages in algebraic geometry, see Blechschmidt's notes [0]. In particular, section 20 is dedicated to proving things that are difficult without the internal language.