I don't think this thread is a major spoiler, but, when someone puts that tag in their post title, I don't change it. You'll recognize my tags - they will look like this [SPOILER] and you'll see that I've edited the post!

Now, on to your housing dilemma! Only a builder can initiate housing construction. You don't say how many puzzles (or which puzzles) you've completed, so I can't say whether or not any construction projects remain for your villagers. If you have, indeed, finished all of the construction projects, there is still the possibility of an Island Event that will provide the opportunity to train a builder.