^^^^^My wife puts our grill in the open shed for winter storage. She covers it and mice build in there every year. I can’t convince her to leave it uncovered.
As for wire made of soy or other product. Hmmm, sounds like we could change that easy enough.
I don't think that it matters if they are covered or not. Last summer I took my little tabletop Weber out of my shed and went camping. I started it up and it just reeked of mouse *.*. It was too dry to have a fire and I only had one disposable tank of gas. I dunked the grill in a stream, cleaned the crap out of it (literally