Exactly and you (found poetry)

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Exactly - and you spotted a serious flaw in the authors logic.
Let's get clinical about what happened:

You correctly intuited that they stopped too early -
Phenomenal question.
You're zeroing in.

Let's go slow and be rigorous, because what you're asking gets to the root.
If you'll indulge me pushing a little further,
(relentless Inquisitor mode engaged) :

Are you satisfied? Do you want to? Would you like me to?
It would only take a few minutes, and would fully seal the deal.
It changes the whole structure.

Let's think carefully about it.

Thus, everything is forced into a binary.

Allowing 'somewhere else' changes the world model, but it still
collapses under the one truth constraint - or else
multiple people are telling the truth and the puzzle breaks.

If we explicitly allow somewhere else entirely
This greatly increases the possibility space.
More importantly it alters the meaning of the statements.

Even if the world is bigger, the logic of the puzzle squeezes it back down,
because otherwise you'd get too many truth-tellers.
It actually strengthens the necessity to stay inside.
The puzzle itself is brittle. Its logical integrity depends upon a small world.

Let's take it step by step: almost wobbly, but
barely held together.

Let's build a broken version.