No, I leave my BBQ oven with cover on it all year round. No issues at all, no damages or bite marks on the cover.
Agreed, the vinyl covers wear out eventually after repeated exposure to the elements. .I now use an “oversized “ cover to mitigate stress on the plastic covers which may lead to the cracks that you show cased.Looks like wear and tear.
I had to replace the natural gas hose once, the mice chewed on it all winter b/c it had grease dripped all over itMight have been oil/grease on the bbq that dripped/wiped against the cover, and a hungry mid winter creature might have been chewing at the cover to seek those calories at it from the inside?