It's hard to say what is normal. Small children don't walk heal to toe, they learn that as they grow. Also, people in parts of the world not using shoes can also preserve a little of the infantile walk, which is kind of natural, you don't want to slam your heal and whole foot into a sharp object. People who do wear shoes very often, usually cannot do anything but infantile walk if made to walk outside. They simply have sensitive feet. Maybe you have overly sensitive feet?

Infantile walk can be preserved mostly in autism, where the brain seems to stay in a less developed stage for much longer. It might hold true for other conditions as well.

As for fast runners, they don't run on their heals at all.