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All unvaccinated international travelers to the United States must take a COVID test; it doesn't matter why they are unvaccinated. If the test comes back positive (or if the traveler develops COVID symptoms after arrival), then they must must self-isolate for 5 days. Self-isolation is done in response to a positive COVID test or COVID symptoms, not speculatively.
Some unvaccinated international travelers additionally need to quarantine on arrival (which basically means they need to self-isolate just in case they have COVID, rather in response to knowing they have COVID). This does not apply to your child.
Also note that, contrary to what you said in your question, the COVID test must be administered 3-5 days after arrival, not "on arrival".