A coworker of mine had some problems to get her son admitted to a top school here in Chile. The kid lives in a bilingual home and speaks both English and Spanish at a level below the level of a monolingual kid.

This looks like a positive data point for your hypothesis.

Not quite. The difference is that bilingualism (which also has a host of other cognitive advantages and has been associated with e.g. being a protective factor against dementia) will lead to both languages being delayed, while sign language will develop even earlier than spoken language, and then delay the appropriation of spoken language.

Also, not any sort of a communication should be categorized as "language" as far as our brains are concerned, we have specialized areas associated e.g. with coordinating spoken speech and mapping concepts to speech, as opposed to hand motions (i.e. Broca's area).