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.
As a self-employed contractor, it's quite common for me to negotiate the contract and wait for screening while already working on the project. They still have to pay me for time worked, and if we can't agree, they're screwed as much as I am, because they're already investing in onboarding, and I haven't delivered anything yet.
So there is some room for negotiation, although some things aren't negotiable for them. I've often had a non-compete clause of some sort in my contract, and I've always been able to get its scope reduced, but mever been able to get it dropped entirely. But if you ask up front about it, it's also to their benefit to be honest about it, because hiring you and then having you quit at the last moment because of their lie is not good for them either
The other option is to get the contract signed, or at least seen, before you quit your previous job. If it's a job for which you need to move and dramatically change your life, then you should definitely get some hard commitments from them before you do anything drastic.
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.
How would that help? Whether it's written or not, the agreement is as you say, until one party decides to end it.
And even with such an agreement, they might still ask you to sign something on day one. If you don't sign it, they may very well decided to "terminate the agreement" on day one.
If you're fired, you're eligible for unemployment. You're also eligible if you quit your job for a new company but then they rescind the offer. That's basically as much protection as you get in the US, since most states are at-will.
How do you avoid this? It's a particular kind of institutional trickery that may not even be intentional but is very frustrating.