The thing about bullet hells is that they are actually super generous. With a classic shooter, you often get shots moving at you super quickly and you basically need to memorize where they turn up, which feels unfair. Bullet hells are super rewarding because it constantly feels like you should be dead, but you make it through. It's not about a bullet overlapping your ship, it's about that one pixel in the middle of the ship hitting the one pixel in the center of the bullet.
So in effect a bullet hell creates a pattern where you need to identify the gaps (or at least the lowest density of bullets) and move towards those, and to avoid actual walls of shots (like beams). There's usually also ways to clear out every bullet, for those "impossible" situations.
Also, most every bullet hell has a way to have or earn unlimited credits so if nothing else you can just credit feed through it all for the experience. There's also instant restarts, not using checkpoints like older games.
I'd recommend on modern systems, if you can find them in your region:
Dodonpachi Resurrection
Mushihimesama
Batsugun (the "first" bullet hell)
Progear (available within Capcom Arcade Stadium)
With the exception of Batsugun, these should only cost you like $10 for a straight port. There's also more ambitious ports like the M2 ShotTriggers, but those are like $40 and have a billion cool features that you don't need.
Ikaruga and Radiant Silvergun are also amazing games, but not as "pure" bullet hells with more puzzle elements. I do like them more, but they may not represent the genre.