Mizubaku Daibouken Liquid Kids is a fun non-shmup arcade
platformer.
Gekibo (Gekisha Boy) isn't anything close to a shmup, and it's pretty interesting. You take pictures of crazy shit when it happens.
NINJA SPIRIT. How do you not already own this? Sidescrolling Ninja action. You may also like Kaze Kiri, but it's not even as good & is a gajillion times as expensive.
Splatterhouse may be basic as far as beat-em-ups go, but at least it's got more going for it than Vigilante, like the incredible atmosphere.
I think everyone should give Samurai Ghost/Genpei Toumaden Kannoni a shot, but it's not for everyone.
If you like driving, there's always Outrun, Chase HQ...I can't comment on Power Drift so I'm not sure it's actually a good game, but it looks interesting.
The last game I'll mention in depth is a shmup of sorts, but you need it anyway: Space Harrier. The turbo/PCE versions are both pretty affordable & it's one of the best console ports. I can actually get farther in it than the 32x version. Anyway, it may be shooty, but there's something about the very basic nature of the gameplay & the whacky enemies you come across that makes me go "just one more play..."
Other games I'd play given the ones you've got already:
Genji Tsuushin Agedama
Son Son 2
Kaizo Chojin Shubibinman 2