Good luck with that one. To put it mildly, the concept is.... not a very popular one. The problem with the idea in the eyes of many fans is that increasing the player's power via progression shatters the difficulty of many things. Same reason I don't like progression in roguelikes actually. An early game boss, normally a threat, loses it's difficulty NOT because you got better at the game, but simply because you have such good stats (that you STARTED with) that you could act like a total fool around it and still win. In a genre that people flock to *because* it is skill based, this... usually doesnt work. It also means that after enough runs, the early parts of the game become.... well... completely pointless. A total waste of time, because nothing can hurt you. Considering that shmups also not only have low focus on story, but often literally *no* story, there's no need to create progression based on that either.
Not to mention that shmups as a whole are designed to be played for score, and score + progression is a huge no-no.
Exploration elements are even more rare. I cant think of any that do that. You can find "hidden" things but they're usually like Dodonpachi's bees, they are there for points and they'll be in the same exact spots every time.
As for the idea of unlocking stuff, ehhh.... usually, on the very rare occaision when this is a thing, all you're unlocking is other ships to play as. There isn't anything else to unlock, you see. Shmups don't really have items or anything to unlock, and there's typically only 5 stages or so (and locking THOSE would just anger everyone).
Hell, even games that are roguelikes + shmups, such as Steredenn, don't do this. Heck, since it was mentioned, Starward doesn't do it either; we have no permanent progression in that game (aside from unlocking the final two floors, but I don't know why that's even there... it probably made sense to someone at the time).
Of all of the shmups I've experienced.... I cant think of even one that fits the bill here. MAYBE Darius (I forget the full title, it's on Steam). Maybe. But even then, for everything that one has, it's probably still not the sort of progression I believe you're looking for.
Typically, if you want the bullet-hell / shmuppy sort of gameplay, but with more than the ultra-straightforward shooting, well.... yeah, you're gonna have to go for games like Isaac or Gungeon or whatever. But even then, like Starward many of those have permanent UNLOCKS, but not permanent "progression" if that makes sense. In Gungeon for instance your characters will never, ever get more inherent power... you'll just have more items that can appear in a run (which could be good, or could suck).
Last thing: I looked up this "Tenta Shooter" to see what you meant by the small screen bit. A warning here: ALL vertically scrolling shmups, and I mean *all* of them, will do this. No, wait, I can think of ONE exception that uses the entire screen, exactly ONE, but I cant remember what it's called. Generally though, for scrolling shmups, what you see with Tenta there is the norm. Exceptions may exist, but they'll be very rare. However, typically you can zoom that middle bit in somewhat, stuff like that. But you're still going to have huge side-bars. The genre's design comes from games that appeared in Japanese arcades that literally had vertically-aligned screens, while here we are with horizontal ones. There is a way around this, but you need a screen capable of rotating. Any of these games can be put into "TATE" mode, I think it's called, which basically rotates the whole thing 90 degrees. But, yeah, you have to have a monitor that is physically capable of that. And even then it can be a bit of a pain (I have one, it can do it, but the process is a little irritating so I often don't).
And yes, I know that whole bit is a little stupid. You'd think that a game of this genre developed for a device that ISNT a freaking arcade machine could, you know, try a different approach with the screen, but.... nooooo, that'd make too much sense, right? Gotta just do it the TRADITIONAL way. Bah. So everyone just does the super-restricted-screen even when the game in question is not an arcade port. I've always thought that was a bit silly, but.... I don't get to make the rules here, do I....
Obviously, horizontal-scoll shmups don't really have this issue..... usually. They sometimes still do.
