Okey I think I grasp where you're coming from.
First on the setting.
I think the main issue here is there is a requirement in the Tv/OVA to pay homage to the original art; I wouldn't be surprised if it isn't even contractual. With the game, not so much. I would say, like the manga, it sells more on the brand name of "Black Rock Shooter" more-so then the art it comes from. Basically, if you're making Tv/OVA you're probably going to be selling figures, artwork, and other merch, and obviously people who already have some sort of BRS stuff would want matching merch to go with it--it helps build the brand up like that.
Getting the BRS art-world is pretty challenging. The game from what I see and you're telling me takes the "chaos reality" approach, and that kind of works, but it's not really the BRS-crazy-level land. The Manga/Tv/OVA use the Soul Space excuse, and small differences aside (Manga/OVA using Dimensional, Tv using Psychic/Psychological) they roughly manga to get that vivid world from the BRS art.
Another problem is how much explanation you want to have. So for example, by setting it in a sci-fi world you got a lot of crap to explain, and you can choose to not explain it but that comes off as cheap. On the other hand by going with fantasy you can always pull the "magical girl" card (OVA), or the afterlife card (Manga) or the imagination card (Tv) and you got a more mold-able and varied power scale. Of course the caveat to this is that the less you explain the more confusing it gets as you move along, because you incur the burden of accepting things for what they are. But IMO that's not such a big issue, at least for fans of the art, because everyone already had a rough idea of how things play out. So hence, nobody say questions why BRS has a big ass cannon, her blue flame eye, and so forth. It's interesting that BRS (and I suppose Touhou kind-of) can get away with it that way.
Going back to the homage issue. It's just about, let's call it fanservice, it's also about avoiding backlash. Basically, even though there's not much of a "franchise" before it, there is a fanbase, and there also is a general conception of what BRS is (from people who just say come across the artwork here and there). If say just take the name and a few elements (black jacket, flaming eye, twin-tail black hair) but change everything else, you're very likely to alienate a lot of people. I'm sure there were plenty of cases of people looking at the game, who were semi-familar with BRS, and going "WTF is this" because just from the cover and artwork it just rubs you the wrong way when it doesn't match what you had in your head BRS to be.
You point out how the game version has a personality, but really that's necessarily a good thing. BRS art reads well because BRS herself just doesn't have a personality, she's basically just the pure representation of a cool looking girl with a gun. The manga does it too BTW, in the manga BRS is dressed like an indian and smiles and has a snake sword instead of a gun as the main weapon. As character description that's not bad, but obviously if you expect BRS, that sounds like some an extreme deviation/look-alike. This "deviation" problem applies to the setting too, people expect "crazy unexplainable world" not "apocalypse" (game) or "purgatory" (manga). I'd say even if you're obviously gonna have people that can accept the deviations (and I'm one of them), it's probably a better experience overall not having the deviations.
Regarding the character cast...
First off I don't find the Tv having any kind of cast size or cast dynamics problem. You basically have Mato, Yuu, Saya, Kagari and their Other World counterparts, and that's about it. Sure you have Mato's mother, brother and some classmates, but considering they don't even have a name (or at least I haven't really heard their name) I would say they're not even side character, just background characters (required by the setting). The characterization is pretty straight forward (at least for everyone but Mato): picture book story > other world counterpart > real world problem.
Now, I'm not sure where you were going with your point, I guess you meant to say that all the background characters were too much and it should all have been a lot more scarce. While I can sympathize, it just doesn't work like that. This isn't just an animation project, it's a Tv series in a particular slot. Unlike the game it's not aimed at hardcore fans, it's aimed at the general public, and just like if it was a live action you would have real people, so here because it a general broadcast you have to have roots in the real world for the story (so as to not alienate a lot of your viewers that drop in). Let's say you didn't know anything about BRS, and the Tv didn't have the real world connection (or a very shallow one), you would be totally lost on what the hell is going on. Don't get me wrong I would watch a show with out or less of the real world stuff, but you know oranges don't grow on the apple tree, so the conception that it-should-be-like that is misplaced in this case.
On the premise...
The premise of the Tv series is "the little bird picture book story". And I guess that's a little more sophisticated but I don't see it as THAT much more sophisticated then the "world ended, go fight aliens". Sure the game version is shorter then the Tv one, but that's only in direct comparison, in a general comparison to other stories out there, it's pretty simplistic:
- bird flies around
- bird collects colors
- bird dies
Of course I don't think the execution isn't ideal, and the story hits some bumps along the way; more notably it hits some sort of climax events before the ended and when it actually goes back to normal it all seems anti-climatic. But, that's not really a fault of the story; you can't say the execution would be better if it was story Y instead of X.