How can I be fair to both the company and its people in HR?

You have set yourself a clear standard: hold both the organization and its people with justice, and never use one as an excuse to harm the other. That is exactly the trap in this work, hiding behind the company to mistreat a person, or behind a person to be reckless with the company. You have named it, which is the first defense against it. Justice here means the same rules apply no matter which side is inconvenient. You do not bend the truth to protect leadership, and you do not bend it to protect a favorite. You look for what is actually fair and you say it, even when neither side wants to hear it. When you make a call, ask whether you are using one party as a shield for harming the other. If you are, stop and find the honest path instead. That single question, asked every time, is how you keep this promise real.