Children traveling by air are required to have a valid passport to depart or enter the US (or certain alternative identification such as a NEXUS card or military ID). There are different rules allowing for use of a birth certificate for land or sea crossings, but a birth certificate alone is not sufficient to travel by air.
It sounds like you got a lot of bad advice along the way, but the agent that denied boarding was ultimately correct.
Originally Posted by
Boghopper
Could it really be true that you could leave the country with your kid and then not be allowed back in? That seems like something UA might want to nip in the bud by not allowing the outbound flight.
The CBP requires a valid travel document to be presented in order to leave the country as well. It's a fairly recent-added requirement (at least if you consider post-9/11 recent).