I think something like Spine might fit the bill, for the animation. You illustrate parts of a character in relatively high def, arms legs torso head, etc. You'd also need to illustrate variants of body parts so that it doesn't succumb to the same fate a lot of Spine/2d skeletal games have, and look too stiff/inorganic.
http://esotericsoftware.com/I dont have any experience in it, but last I checked it works with GMS2.
Spine only takes care of the animation though. Illustrations pre-animation and which tool you use to make them is up to you. I think games like Spelunky use a combination of vector art and digital painting. I think Darkest Dungeon also used Spine (or something similar).
Cuphead is completely different, as you say. I think I saw a documentary where the animators didn't even use software half the time, but actual traditional onion skinning, and a legit rotating animator's table and lightbox.