Maybe this is different here in Europe, but HR's role is to protect the company. HR in my experience does not do whatever a manager wants. Let's say you have a toxic direct, unless you have a 50 page document outlining every single transgression or bad behavior and the steps you took to mediate said behaviour, including corrective feedback, and statements by the toxic employee's peers supporting your observations, HR will often times do exactly nothing and happily ignore you, and even then, they could still propose "mediation" or "conflict resolution" procedures because they are afraid of an unfair dismissal law suit or whatever.
I find this attitude to be the same as kids not telling on the bully because it’ll make things worse. The guy lost a four figure bonus for no sane reason. Keeping quiet isn’t going to make anything better. You either speak with HR or firstly maybe your boss’s boss. Maybe things are a bit different if you work in one of the places where you can be fired at will but generally you have employment rights that will protect you.
Never talk to the bosses boss without going to your boss first. By doing so, you are ignoring the chain of command and likely will you fired. How would you feel if someone you managed went behind your back to talk to your boss?
HR's exact goals vary between companies, but unless the company culture is toxic all the way to the top, part of their job is usually to reduce staff turnover (which is very expensive for the company), and in pretty much any company their job is to protect the company against lawsuits. Abusive managers are not well liked by HR.
That doesn't align with my experience. In my experience, they pretend to care about the employee, but when the employer is in the wrong, even in an obvious way, they will side with the employer in the blink of an eye every time instead of trying to reason with the employer.
The employer, yes. Some low-level manager? Not at all. It is not in the employer's interest to take the side of a toxic manager who may cause several of their reports to quit.
Just as you have to earn the respect from everyone, each single individual in that org must earn your respect. Choose your confidants carefully. HR is for door in, and out.