Could be a little bit simplistic I think. Yes basically company has to come first definitely, and sometimes that does mean being harsh, and in cases of redundancy etc there is no middle ground.
However that doesn't necessarily mean HR can't also be somewhere employees go to for help and advice, and sometimes there might be a bit of middle ground.
You would never advise anything that goes directly against the interests of the company of course. Hmm trying to think of an example!
You mention if someone wants more money. Obvious quick and easy answer in interests of company would be to come up with a reason for saying 'no'.
But a manager might not know about other comparable salaries within the organisation, or going rates for that role, and HR might take a longer term view and persuade the manager that granting the request might be in the longer term best interests of the company, or might avoid a problem later.
So ultimately best interests of the organisation come first, but (a good) HR team can also give good quality advice and support to the employees without jeopardising that.
Occasionally that means subtly talking the employee round so that they think what is in the best interests of the organisation was their idea! (shh!) Or it means just listening. This way you are giving advice and support while sometimes massaging to achieve best outcome, rather than just saying yes or no to requests. Also encouraging a culture where e'ees are comfortable coming to HR for support can be beneficial to the organisation as HR then have more knowledge about what is going on and advice to management is more informed, and also this way can avoid use of formal grievances etc if there is an informal way of resolving a problem by advising an employee how to achieve that.
Also, a good and influential (ie not all)HR team want to achieve best practice within the organisation they work in. This might mean advising management to implement policies or practices that cost money, or time, i.e.not directly in best interests of organisation in a black and white sense.
All this obviously depends on the culture of the organisation, how influential HR is, (often not very), and how open the organisation are to longer term suggestions.