Earlier quoted context omitted.
I've been sprung with these before. Usually trivial, unenforceable issues, but annoying none the less. The worst was I once even had a background check _after_ accepting and signing. Nothing came up, but that could have left me with no job and no recourse. How do you avoid this? It's a particular kind of institutional trickery that may not even be intentional but is very frustrating.
In the US employment is at-will, so it's sort of a moot point. Sure they can fire you because of something on your background check, but they could also fire you 3 days after starting because the manager doesn't like you. Short of a serious felony conviction it's unlikely to come up, and if it does discuss it or move on to thousand of other companies.
And yes you can move on, but you're still in an unnecessary position (looking for work while not employed) through no fault of your own.