A result like Dirichlet's theorem makes sense to me as worthy of attention because arithmetic progressions are a natural mathematical phenomenon. But then there are results like the Friedlander-Iwaniec theorem which states the infinitude of primes of the form $a^2+b^4,$ and a theorem of Heath-Brown which states the infinitude of primes of the form $x^3 +2y^3.$
I don't understand why top-notch mathematicians pursue these questions, let alone receive prizes for them ("Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work." according to the linked Wikipedia article). It would be one thing if the primes were of a more natural form like in Dirichlet's theorem or applied to a broad class of forms, but these seem to be stand-alone results for very specific, unnatural forms. Why are these questions researched and and their solutions celebrated? I don't have the necessary sieve-theoretic background to understand the proofs, but is it the case that these theorems are just corollaries of much deeper and significant theories? Or is the motivation something else?