Actually, "their" for "his or her" is not grammatical.
"Rule 11. The use of they and their with singular pronouns is frowned upon by many traditionalists. To be consistent, it is a good practice to try to avoid they and its variants (e.g., them, their, themselves) with previously singular nouns or pronouns.
Not consistent: Someone has to do it, and they have to do it well.
The problem is that someone is singular, but they is plural. If we change they to he or she, we get a rather clumsy sentence, even if it is technically correct.
Technically correct: Someone has to do it, and he or she has to do it well.
Replacing an inconsistent sentence with a poorly written one is a bad bargain. The better option is to rewrite.
Rewritten: Someone has to do it, and has to do it well."
Whichever language, native speakers tend to rape it.
https://public.oed.com/blog/a-brief-history-of-singular-they/ <-- they was used as a singular for 400 years at least until it became "wrong" in the 1800, around the same time english suddenly made up of its other stupid rules, like not ending sentences with prepositions. The singular they MAY have even originated before the plural version. Its not the speakers screwing up language, its the people who think they can just drop new rules on people and theyll b followed. Languages evolve best naturally, and rarely by artificial selection.
In business and educational settings in the US, the practice of using "they" and "their" is now encouraged in recognition of the LGBTQ community, to avoid calling someone "he" if they identify as a girl, for example. Journalists are also doing this now, so I think it is becoming acceptable through usage.
You see this a lot prior to the 1970s where he/his/him was the prefered generic pronoun when the gender was intended to be all inclusive. Despite this, "they" has actually been used with varying degrees of acceptance since the 14th century as a generic epicene pronoun. That being said, "they" has definitely taken prominence as the epicene singular for the past 40 years and continues to become more prevalent.
I have seen the arguments against it as you mentioned that it does not utilize the correct conjugation of to be for a singular noun.. no one is saying "they is" and i doubt we shall see that anytime soon. I personally subscribe to the idea that language serves culture and that "they" accomplishes a certain social function that has become foundational to our society. "They" has become the ideal choice for a generic singular epicene pronoun because of its inherent all inclusiveness and because it had no inclination to the more patriarchal aspects English. You're definitely seeing a lot more opinions in support of it and it's made enough penetration into our culture for me to consider it grammatically valid.
Good question. Italian's suo works for "his" or "her" (or indeed "its").
"His or her" is only used in English if we are not adept enough to form an alternative phrase. We do have a neutral possessive pronoun: "their" can be used as either singular or plural if there's no ambiguity as to which applies. We can switch to the impersonal "one's" if the subject has already been made impersonal by "one".