In the USA where most states are "at will" employment, a "PIP" (Performance Improvement Plan) or providing multiple chances for an employee to become better isn't required by law: https://www.google.com/search?q=employee+%22performance+impr...
That said, some companies in some industries add a PIP as a social norm rather than a requirement of law.
E.g., An underperforming programmer might be put on a PIP. However, in more cutthroat industries... a new car salesman who isn't selling the expected quota of cars or a hedge fund associate who lost several million on a bad stock pick are often fired immediately without any PIP.
I suspect they often do the PIP as legal protection, documentation they can produce later if needed. Even in an "at-will" environment, employees can sue for wrongful termination.
Employers in the US don't care about giving people the opportunity to improve, it's more about documenting the underperformance so there is an airtight case against the employee - nobody can claim a protected reason if the underperformance is very well documented.
All of this said, and this applies to smaller companies more than bigger companies due to the risks involved, if the employer is very confident the employee won't be able to argue that they were fired due to a protected reason, in most states they can just fire them in the middle of the day with no justification: "I don't like you, out of here".
Yes, there are cases where the employee needs to go. But there are others where that's not exactly true.
All that said, it's a relationship. Life is too short for unhealthy and unproductive relationships. When I find nyself in such situations I update my CV and start looking to move on. Somethings are what they are and not worth the fight.
Actually, these efforts often don't help much. As a rule, if you find yourself doing elaborate cover-your-ass excessive, you probably are doing something very systematically wrong. I've seen plenty of situations where the well-meaning managers documented everything, did the whole document poor performance, and a single deposition of a manager made it clear that the entire process used to document and remedy under-performance was simply collusion to fire someone for a protected reason... even though in reality, it was not. It goes kind of like this:
"So after four years of not a single entry in ___'s file, suddenly a poor review and a complicated PIP?" "____ just wasn't fitting in on the team."
"Hmm. _____ is the only _____ on the team, despite _____'s race/religion being ___% of the population. All hires made after you became manager were (insert manager's race/religion/sex). No other employees on your team are on PIPs. So, why is it that ___ wasn't fitting in?"
"I can haz consult with lawyer now?" (followed by settlement for 2x manager's salary)
Doesn't really matter. The problem is the choice for the employer is really:
A risk of paying 3 x whatever (sometimes crazy) number the employee is asking for
versus
Certainty of paying 1 x whatever we settle out of court for
Risk aversion kicks in, and so the company settles.
You typically need to give the under-performing employee at least a chance to improve and meet the required standard - otherwise the hiring company will be leaving themselves open to a wrongful dismissal suit.