The simple fact is, Pat, that language isn't 'invented' in this way; language evolves through usage; like it or not, more and more, 'they / their / them' are becoming the 'androgynous' singular pronouns.
There's nothing unusual, incidentally, about a pronoun 'multi-tasking' in this way: 'sie' in German can be 'she', 'they' or 'you' (singular and plural).
I agree with you, however, about using 'they / their' for the next pope: the next pope must be a man, the legendary Pope Joan notwithstanding, so 'he' and 'him' should be used here (imo).
MK