With respect to the Hyatt and other chains, this is not the kind of resort my wife and I prefer. Think Ladera Resort in St. Lucia or Parrot Cay in the Turks & Caicos, we prefer smaller, higher quality "resorts", preferably with spa amenities - on the beach is nice, but secondary (the island seems small enough to drive wherever we want). These 40 year olds prefer high quality over quantity (rooms, people, etc.).
Staying for a week without email, telephone and our (adorable) children!!, we would prefer quiet, low key bliss. So maybe no such accommodations exist in Kauai, but I thought we'd ask those more informed.