I've never heard of a PIP actually succeeding (an employee improving to the satisfaction of their manager) so it delays the inevitable, perhaps allowing the employee to quit and find another job before their fired. It's definitely meant to avoid a wrongful dismissal lawsuit but I think that almost never works; either the employee is going to sue or they're not, "being given a chance to improve" isn't going to change that.
If there are explicit goals or metrics to be measured, which I have seen in many cases, it's pretty clear what needs to be done to clear it. If they don't meet those goals, and they're attainable (e.g. other people in the same role are performing that well on average), it's a pretty clear defense for a lawsuit.
The whole place was a clown show that I was unhappy at anyway, so while I entertained the idea of taking them to court, when I realized the PIP is basically an elaborate attempt to defend against wrongful termination cases I ultimately just found a much better job.
The biggest thing is to ensure that the targets in the PIP are both realistic and measurable. All too often I've seen people being asked to meet a bar far higher than is possible (and not even met by top achievers at a company) and with no real way to measure it other than "if I think you're doing good enough".
Yeah, it's a silly game where everyone has to pretend there's a chance you can "succeed". When in reality it's "rather than firing you and dealing with severance and lawsuits, we'll keep paying you for a few months while you find a new job, and then we'll agree that you voluntarily resigned". An employee who doesn't realize that is going to have a bad time, and would in fact be better off fired.
Side note since you mentioned "firing". The article's title ("facing firing") may unintentionally overstate what Amazon's appeal process is actually appealing.
Based on the text, Amazon's "appeal process" is to challenge the status of being put on PIP - the Performance Improvement Plan. It isn't to appeal a firing. (Although I understand that some see that as no difference if a PIP is ~99% equivalent to getting fired eventually.)
If the employee wants to keep working (at least temporarily) for Amazon, he/she doesn't need to appeal. What a successful appeal does is let them work without PIP targets hanging over their head.