Your second point is wrong. Obviously it narrows specificity by half the room on average. I don't know why you'd argue against that obvious fact.
It's basic math: If you have a set C that is the union of two sets A and B where A and B have the same cardinality, referring to "a C" gives you twice the possibilities than referring to either "an A" or "a B." So it's measurably twice as efficient to do the former. Since so many unrelated languages in the world ended up with such a system (or very close), it's reasonable to think that that efficiency was worth it. Since most of those systems are not much more specific, it's reasonable to think that being more specific wasn't worth it (one can always specify further using more words.)
Of course it narrows down specificity. My point is that in today's society, it seems mostly useless to me. Most of the time I don't need to narrow it down this way at all. This may have been different in a world where people were segregated by sex so intensively that half of the population didn't even have the right to vote, but today I fail to see the usefulness of it.
Then you fail to see the usefulness of specificity and efficiency in speech, which is both weird and explains why it took you so many words to say that.
If I'll want to refer to you and this conversation when talking to someone else, I don't need to refer to your gender at all. Just like I don't need to refer to your race, your social class or color of your hair. Stopping to consider whether I should use "he" or "she" (or maybe something else) is the exact opposite of efficiency in speech.
It seems to me that it's actually you who misunderstands the usefulness of specificity. It's not useful to be overspecific.
Obviously it depends on your goals and the context. Both specificity and generality are useful. To ban one is foolishness. They coalesced into short words for a reason: people use them, a lot.
We can exchange truisms all day ;) But that doesn't change the fact that in my experience specificity related to gender pronouns is needed (or even helpful) only in a very tiny minority of everyday contexts.
For the record, my native tongue is much more gendered than English (it has gendered nouns, verbs and adjectives; not just pronouns) - I don't understand how it's useful at all, I don't miss it in English.