Harry Clearwater shows up more in 'New Moon' and 'Eclipse', but he's never really front and center. He's one of the elders on the Quileute tribal council alongside Billy Black, so he's part of that whole decision-making group when it comes to the treaty with the Cullens and the werewolf history. His daughter is Leah, who's the first female werewolf, and his son is Seth, who imprints later on. Honestly, his biggest role is his death. He dies from a heart attack when the newborn vampire army is approaching in 'Eclipse', and that death is what finally pushes Jacob to tell Bella about the tribal legends and the real reason he pulled away from her. It's a catalyst moment, but as a character, he's more of a plot device—a nice guy who shows the human cost of everything happening.
Thinking about it, Stephenie Meyer uses characters like Harry and Sue Clearwater to build out that Forks community feel and the Quileute family networks. Without Harry, you don't get Seth’s whole sweet, optimistic personality as clearly, or Leah’s deeper bitterness which is partly tied to losing her dad. The books lean on those off-page relationships to give weight to the on-page drama. Harry’s role is functional, but it works because his absence creates ripples. His death also makes Charlie Swan’s fear for Bella way more tangible, since Charlie was Harry’s close friend. It all connects back to making the human characters feel vulnerable in a supernatural conflict, even if Harry himself wasn’t super fleshed out.