If you never played super air wolf/crossfire on the megadrive/genesis then not necessarily what you are after but could be scratch the itch after a fashion.
Likewise I am probably supposed to point you at the Nier franchise. Most of its bosses seem to go for shmup style gameplay, indeed bullet hell, after a fashion but the rest is a nice RPG.
Back on scratching them same itch then might want to look at twin stick shooters. Many of the modern ones have at least broached such topics. Few roguelikes/roguelites also went there -- PS4 PSN+ in Europe had one that did something for that (future sci fi type deal, you are taking down a corporation by waking up literal sleeping agents and sending them in). Sadly I don't have anything in the likes of "like Rogue Legacy but shmup".
For some earlier stuff then the DSiware games, most of which got backported to the DS, did a small shooting robot crawling around and exploring type thing that is kind of where that was going. Though one that was not and might be worth investigating is Dark Void Zero (a handheld spinoff of the less than stellar jet pack game Dark Void from home consoles in a similar timeframe).
You may even want to go even more old school. Some of the gravity games that once sat alongside shmups. https://www.turboraketti.org/tr/index_e.html
and Thrust for the Amiga tending to be the main ones. Though if I am doing Amiga and this sort of thing I might actually suggest an old favourite in Simulcra, more on the metroidvania aspect (though far smaller "levels") but still in a ship and some reactions needed.
Sigma Star Saga mentioned above is a good game but not sure it would go for this. If we are stretching things though then do make sure you played megaman battle network (5 on the DS is probably my suggested entry point) and can go back and do some GBA things. Its battles scratch my reactions, patterns and whatnot from shmups itch.