Hot take : Alien 3 is not a very good movie no matter what. Not only it lacks a McTiernan-like sense of space in the labyrinth chases, but it slso make a big fuss about whether Bishop is an android or a person, which actually doesn't matter at all (and actually flip-flopped depending on the drafts). And this is without even starting on the clunky "i would prefer if Aliens had ended differently wait let me correct it quickly" premise.
Hot take: Alien 3 is criminally underrated.
Also, I'm not sure about some of your points. The film doesn't make a big fuss as to whether Michael Bishop is an android or not - sure, he himself does, but only for a few minutes, and it's beside the point.
I also don't know what you mean by the "correction" premise - Alien 3 flows on from Aliens, it's not "correcting" anything, unless you mean that Hicks and Newt are dead before the film properly starts (which is a complaint I've never understood - it seems to be based on a fundamental misunderstanding as to what the Alien universe is actually about*). If anything, more people have tried to "correct" Alien 3 than vice versa, whether it be Colonial Marines (altering Hicks's death), Blokamp's attempted Alien film, or the Alien 3 unproduced screenplay novel/graphic novel, both of which were drek. It's telling that every attempt to "correct" Alien 3 has been worse than the film itself, or in Blokamp's case, thankfully never made.
*As in, grimdark - good, decent people die horribly, life sucks, and over the course of history, it gets worse. Having Newt and Hicks die is perfectly in keeping with the type of universe this is, and certainly the type of film Alien 3 is. It's not even the only time th
Alien 3 was such a huge waste of potential. 3 was supposed to take place on Earth, and apparently the studio was too cheap I and lazy to expand on that scope. Just play Alien vs. Predator (Capcom). It's based off the original draft when the AVP film was supposed to be released in 1994, and has better use of the Xenos are on Earth and wrecking shit in all of Los Angeles.
Alien 3 taking place on Earth was one possible route, it was never the only option. There's numerous Alien 3 ideas that were pitched, including the Wooden Planet concept, and the Anchorpoint/UPP plotline that got adapted into the above-mentioned works. I also don't buy the idea that it's a question of being "too lazy," and even if it was, personally, the idea of bringing the xenomorphs to Earth is one that I completely disagree with for a variety of reasons. And yes, I know that the xenomorphs did end up reaching Earth in EU material (e.g. the Earth War Trilogy), but Alien canon is a mess outside the films, and even if it wasn't, it's academic to Alien 3 itself.
As for the afforementioned game...okay, be frank, can you honestly see that fitting in with the Alien IP, at least at that point in the timeline? You have cybernetically enhanced soldiers fighting alongside yautja against hordes of xenomorphs in broad daylight, in a setting that, apart from a few exceptions, has operated on the principle of "less is more?" Even Aliens still confines its scope to a single colony with a single platoon against a few hundred xenomorphs, with the number of xenomorphs seen simultaniously never reaching digits. Maybe the premise would have made for a better AvP movie, but as a better Alien 3? Can't see that happening.