...take an existing graphic, like the one attached here, and smooth/flatten out the colors?
For example, Bugs Bunny shows a lot of splotchiness on his arms. If those could be
reduced down to a single, smooth, consistent color—without needing me to spend a lot
of time reworking it manually—that would be nice.

I guess I want some type of tool that "averages" out adjacent pixels in large areas, and
then selects one color for the whole lot of them.

You could try selecting by color range and then replacing.

Or you could minimize the amount of colors on the pallete.