I don't know if it's the audio or is it's just typical Greek, but it's difficult to hear the σ at the end of της in the normal speed version.
so if a word like σύζυγος acquires secondary stress with a possessive enclitic (e.g. η σύζυγός του), why doesn't όνομα (i.e. όνομά της instead of όνομα της)?
Της here is not possessive, it's the definite article in the genitive case and gets grouped with γυναίκας, not όνομα. It would be the possessive pronoun if no noun followed.
Yes, I realized that as soon as I posted the comment and tried to delete it but I guess it didn't. Thanks anyways, though; the moderators hear are so helpful!