I take mentoring others seriously, and although I'm not naturally a very patient person, I am always patient with my coworkers and clients.

This is what I mean. You have to have "people's skills" because it is important in a job where teamwork and mentoring are required. You don't have to socialise on every company events or be friends with coworkers outside of work. But being professional and friendly goes a long way. I know of people who may be good at their jobs on technical level, but are bad with managing people because they lack both social skills and social intelligence, which makes them pass over for promotions.