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- A work of art contains political and social messages, not in the least because it's impossible for any author to wholly avoid coloring their work with political or social viewpoints. The author's experience and ideas do not exist in a vacuum, and will necessarily be a product of their time and environment.
vs.
- A work of art was primarily conceived to push specific political messages, and the actual "art" takes a clear back seat to the specific moral, political, or social messages being conveyed.
vs.
- A work of art cannot get funding unless it bends the knee to an outside group of ideologues who modify the work of art so that it specially conforms to the "correct" social values.
I genuinely can't think of an example of #2 that goes the direction you want. Because in the wider context I think you imagine that one trans character in the new harry potter game to be an example but she doesn't exist to push a political message, but because the creators just wanted to have some representation and piss JK Rowling off. But shit like Call of Duty or Top Gun where it's literally military propaganda making the armed forces look cool to increase recruitment is the strongest #2 that's out there. But if you intend any story that has a moral to be an example then we're back to "that's how stories work," we like art that has a point.
So this is a long way to say that I completely agree with you, I just think that most of the common examples of #2 "gamers" get mad at is really #1 where creators are just artists in a liberal environment and are inspired by a new archetypes of people they can make characters with or just want to see their own experience represented. I think it just feels one sided because one of the core attitudes of half the political spectrum is saying how useless art is and how not worthwhile it is as a career so almost every artist is liberal (source: my partner went to art school, I'm a red-blooded conservative compared to them goddamn).
For #2, I think it's a lot more of a grey area. In principle there's nothing wrong with a piece of art pushing its own social and political agenda, but it can become oppressive if it becomes too widespread or forceful. I can imagine that any era prior to the late 90s, the refusal to show anything other than heterosexual and traditional relationships on screen must have been frustrating for some folks, even if they did not have any issue with traditional heterosexual relationships in principle. The same could be true now: an audience could exhaust on having the same moral message crammed down their throat, even if there's nothing particularly wrong with the given moral message in piece of art when taken as a singular instance. A non-social and non-political version of this might be similar to super hero movie fatigue. The first run-up of Marvel films was a lot of fun, but after another decade of them and 60 more films (or whatever it's been) I'm much less tolerant of each new entry, even if I would have enjoyed prior to becoming so over-saturated. And, as I tried to outline in my initial comment, I think there are better and worse ways to go about this. Something like Pan's Labyrinth is aggressively anti-fascist and pro-feminist, but it's universally loved because it didn't put its message above its art, and didn't cram its message down its viewer's throats. It did not alienate its viewers by attempting to winning culture war points, but instead told a beautiful story.
Lastly, I'll say that I agree with a lot of your points, and I think hiding behind BOTH SIDES of this debate is an elephant in the room: people are leaning on principles, but often just find the speech itself to be objectionable. In other words, if they agreed with the speech then scenarios #2 and #3 wouldn't bother so many people as much. I think this is sort of unavoidable when it comes to contentious cultural topics, but I've done my best to stick to the "principles" side of things here. It's important to be open minded here, and understand that the free speech principle (eg: an artist is free to create any work they want) should be separate from my perception of the art. (eg: I really hate this piece of art because of the social values present in the art.)
#2 has also been a thing basically forever, from "look how great $REIGNING_MONARCH is" to "hey look how screwed up our system for taking care of orphans is" (Oliver twist) to 1984, etc.
The ship on changing the pendulum back to the average has sailed. A lot of people who’d normally be amicable to listening have been run off by the impossible constantly changing moral standards of the group pushing 3. It’s gone from “good to know” to propagandizing. People just want entertainment that doesn’t make them feel horrible. People want to be entertained without the feeling certain aspects of some movie/video game/music only exist to “convey the message”.