It's only a matter of finding the right red with the correct undertone. There are red lipsticks with an orange undertone, blue undertone, and pink undertone. It's only a matter of trying them until you find the right shade.
My first tube of red is a beat up old Fashion Fair lipstick ( I treasure it though). I was doing makeup and I asked my mom if she had any red. She disappeared and came back with this:
It's the old school fashion fair lipstick and this tube is probably about twenty years old (and yes I know your supposed to throw it out). I tried it on and fell in love! It was so pigmented and creamy and the perfect shade of red. And I'm convinced it would work on everyone. My mom, my sister, and I all have different skin tones but check it out!