Would it hurt my application if my essay did not focus on (or even mention) my extracurricular activities?
No. There is no rule that your essay has to be about an EC. Some students may use an EC experience for their essay, but many students will use one of a myriad of life experiences (shopping at Costco, riding the subway, a grandparent reading to them, etc, etc) as the basis for their essays.
No. I know a kid who wrote his Stanford essay about playing Legos… recreating famous battles. He was a history buff, but was applying in math/economics. He got in; someone on the adcom actually told him later how much they enjoyed the essay.