Got got all of all types. What a surprise for sure.
1) Grammatically, that's not true. "Somebody left their umbrella in the office. Would they please collect it?" or "But a journalist should not be forced to reveal their sources." [source] https://en.wikipedia.org/wiki/Singular_they
2) My youngest child is non-binary trans. They chose "they" as a pronoun for a number of reasons, not the least of them was that the singular they already exists and is valid. When my child came out, I was thrilled to support their journey. The ONLY issue we had was the plethora of pronouns. Language is about a common understanding so that our vocal and gutteral utterances have meaning. Of course, language changes, grows and shrinks over time and other than with technology or tragedy, it takes time for new changes in language to permeate a society.
Heck I still say that if Klingon can be purely invented, then linguists working with those in the trans community can work out the language barriers. As it is, what was thought to be the most difficult, Spanish due to it's very binary nature and virtually everything being assigned a male or female attachment, has made significant strides in this regard. Latinx as opposed to latina or latino being a primary example.
It's just a matter of doing the work.
Also, many people just aren't aware that it is common to substitute a generic "he" for singular "they" (in matriarchal societies, the opposite happens and generic "she" replaces the singular "they"). Since it's not all that common to question how much is patriarchal in our society, these things seem fixed and standard when they are absolutely choices that are made in society and reflected in language.
And there are instances where it doesn't make any sense or sounds wrong. You just made use of one. On your second point, it's crystal clear that you're refering to a singular individual, but you follow that phrase with a plural one which, personally, sounds wrong.
I will not modify the way i employ language to cater to people who are on a crusade agaisnt society because they've deemed don't like the social norms that are commonly associated with their sex/gender.
Let me put it this way: if one is born a male or female, i will use the respective pronouns (he or she). I don't really care for the social rants.
For two years I studied history at an HBCU before transferring to a state uni to finish my bachelor's. At the HBCU many of the profs and students were right wingers and were only Democrats because of race and the fact being a black republican is often frowned upon in the black community.
Didn't really have any politically charged professors back in university. Even the core class that I did called "Contemporary issues in Law and Society" presented a fairly balanced view of the world and invited people to debate partisan and bipartisan issues.
So were they progressive? I guess so, seeing as a lack of bias signals to me that the person is constructive.