In a previous Away Message, I explained my use of third person plural for indefinite singulars. Let third person singular also be used alternatively; as the gendered pronoun in place of a neuter indicates familiarity, so let a neuter used in place of a gendered indicates a diminutive.
In the instance of animals, the third singular has ambiguity: an animal has natural gender, but lacks the type of linguistic sentience which allows first or second singular forms without anthropomorphizing. Thus,, let the third singular neuter be used.
A neutered pet is just a double coincidence.