Does a mixer exist with real buttons for scene selection?

Context: I "run sound" for my band. By that, I mean I have a mixer next to me on stage where I pull up fx, mute unused mics and bring up mics as needed for band members who don't often sing. There can't be any fancier moves because I am also playing keyboards and singing at the same time.

I've always used an analog mixer because it lets me see everything at a glance, but I'm envious of digital mixers with their compressors and gates and fully-parametric EQs. I'd love to be able to set up scenes/presets for frequently-used mix and fx configurations, but I can't deal with touchscreens (or worse, a tablet or laptop) while performing.

I dream of a digital mixer with all that cool digital mixer stuff, but with actual buttons for selecting presets. Like I can do with my lighting controller (yes, I have that duty, too). Does such a product exist? Or can anyone suggest another solution, e.g. using MIDI?