Can you get any of the Capcom Classics Collection volume for playing onto your xBox ?
Right now, I am figuring you might enjoy playing the original Strider, present in vol. II., but it hardly fits in the "mindless blasting" category.
I'll add KotoR to that short list, if you had forget how good it is.
I agree with Virtua Cop being a nice game, and shooting from the PC with mouse clicks a treat. Though MAME drivers for this one do require some pretty seriously pumping CPU & GPU chips. If you're not completely happy owning a wii and a xbox, may be you could do with a PS1, real or emulated.