I'll give you my more pedestrian reason. The proofs of existence of injective resolutions require the axiom of choice, in one form or another. Translation: these proofs are not constructive, so there are no general algorithms for producing such objects. This becomes a painful issue in concrete situation. This has similarities with another famous existence result, the Hahn-Banach theorem which postulates the existence of continuous linear functionals with certain properties. It is particularly useful for existence theorems for PDE's. Unfortunately it gives you no guide for finding those solutions.

I'll give you my more pedestrian reason. The proofs of existence of injective resolutions require the axiom of choice, in one form or another. Translation: these proofs are not constructive, so there are no general algorithms for producing such objects. This becomes a painful issue in concrete situations. This has similarities with another famous existence result, the Hahn-Banach theorem which postulates the existence of continuous linear functionals with certain properties. It is particularly useful for existence theorems for PDE's. Unfortunately it gives you no guide for finding those solutions.