I love Hatcher's book so I can't disagree with you. (Although he does have a section on category theory in chapter 2, but never revisits it again). But doing anything with sheaves and schemes without categories would feel a lot like doing linear algebra without matrix notation and just using lots of double sums. I also have a copy of JP May's a Concise course in Algebraic Topology next to me and I also love the presentation there. (Although I would absolutely never want to use that book as a first course haha. Hatcher is a classic for a reason.)

Being slightly facetious here as obviously Griffiths and Harris uses sheaves extensively and no category theory.