'kay so I'm writing this WAY too late at night but here's some Dragon Striker headcanons (more or less supported by the show itself, I'm filling in some holes. Also excuse any spelling mistakes and names i don't know, i should really be in bed rn):
Key mentions "some of the best players in the world come from Kal Asteroc" -> there are other schools like Kal Asteroc
Kal Asteroc IS very much a school, but with a hard focus on Gorotama and the banner teams. This does not, by default, mean that all schools have banner teams, but they COULD have.
The banner teams are fucking ancient apparently. I THINK I remember the roses coach saying the knights weren't a thing for 40 years, and the bards coach very explicitly states "the sunspears have been playing for the bards for generations!" to his nephew, which does imply at least, like, maybe about 200yrs? Roughly.
Gorotama IS Kal Asterocs biggest focus, obv, but it's not the only thing taught at Kal Asteroc. It can't be, because we know that in order to become a banner player, you have to try out, which you have to sign up for, and not every student HAS to sign up for it. In the first episode, Milo mentions that he might go for swimming instead of Gorotama, so there's at least a swimming club at the school, even if it's not as big a deal as Gorotama is.
Sugoi insists that, in order for Key to master his Tama, he has to go to Kal Asteroc. She doesn't explicitly say that he MUST be banner player in order to master his Tama in that same moment, though she does mention his destiny has the dragon striker after he unlocked (?) his Tama.
This leaves the question whether or not Kal Asteroc, while heavily focusing on Gorotama, is also just supposed to teach its' students control over their tamas. If used right and in control, they can be very powerful, so what happens if used wrong or not in control? We have what happens with Key's tama, but he seems to be an exception. Do all tamas go out of control like that if suppressed?
This leaves another question unanswered if every person has a tama or not. Key wasn't shown *sad* at not having a tama before he unlocked his, and his dad (as well as Ssyelles dad) were not shown using one, as far as i could tell. It's possible that using one's tama just isn't common outside of specialized cases (like Gorotama or maybe other occupations), and may even be forbidden in public spaces/highly populated areas.
Back to the school, Kal Asteroc has a gym, which, if it's sole focus was the banner teams and Gorotama, wouldn't make that much sense to have, unless there were other people (read: non-banner players) using it. I think the gym also doesn't have integrated goals like the different fields of the banner teams.
While it's not shown or even mentioned much in the show, there have to be classes going on at Kal Asteroc, right? Like, non-Gorotama-related classes like, idk, math and literature, etc.? I kinda wonder if banner players just don't have to attend those, or if it's just straight up skipped in the show due to a lack of time to tell the story. (remember when the average cartoon season was about 25 episodes? I miss those times...)
Also, if there weren't, Kal Asteroc would literally only be housing the banner players, the coaches, and any staff they might need. Just the banner players (with 5 players to each team) makes 25, plus the swap-in players (idk what they're called) let's make that, idk, maybe 2-3 per banner team (except for the knights which we know don't have any swap-ins yet), so about 35, plus the four coaches, 39.
That whole campus and extra random students tell me there's a lot more than 39 ppl attending Kal Asteroc, namely the knights fan club, none of which are banner players, I'm very certain. So there HAS to be more than JUST Gorotama and the banner teams going on at the school. (I know i'm repeating myself, I'm just trying to imagine what happens in-between what we see, y'know?)
In the gondola ride to Kal Asteroc, as well as some other shots, we see a variety of animal-mixtures (like the donkey-bugs, but also what looks like dragonfly-frogs, there's some weird kind of bird in the knights clubhouse that Ameline takes a liking to, those white birds with colorful patterns, one of which Key pets during the gondola ride as it flies by), so i can imagine some kind of wildlife club either taking care of them, or even just keeping an eye on them, keeping count, checking for diseases, etc... (I like animals)
Also with regards to the donkeybugs, they must have some on campus, because that's what Ssyelle, Key and Ameline shovel off their field before playing for it against the roses (who call it their "favourite compost heap"), and I'm just saying, all that stuff that's composting has to come from somewhere, right? Unless Kal Asteroc, like... imports it? But i feel like it would be much handier to just have some kind of stable on campus. Maybe those animals are also used for lessons or extra-curriculars or something.
Anyway that's it for now, I gotta go to bed, goodnight <3