You suspect wrong. The worker that didn't wash his/her hands after latrine call and sticks them in the ice machine may have put the exact same pathogens in the ice machine that doesn't get cleaned as often as the toilet bowl.
Ice machines are petri dishes for bacteria. I once cleaned one that had a pink slime on the floor of it, when I got to the bottom. It had NEVER been cleaned.
/johnny
People forget cabinet doors, too.
I know folks who had spotless countertops, but their cabinet doors (inside and out, along the edges, where hands reach to open them) were crusty.
If you aren't giving those cabinet doors a thorough cleaning once a week, you're just spreading around the "germ" love.