'Am I right in thinking that in America every child born there has a right to be American ? If so, or in any case, maybe the same should apply here.'
Yes, everyone who is born on US soil is an American by birth. This does not, however, entitle anyone to free healthcare or for their parents to have the right to stay there. Some states offer healthcare provision for children on a sliding income scale. You have to pay for it and/or qualify for it.
They will treat in emergencies, but you will be billed and they will chase up that bill.
That's the system there.
The system here is different. Citizenship operates differently, and entitlement to benefits and NHS operate differently (people who are here on work permits, as full-time students, spouses, fiances and other limited-stay visas do not have recourse to public funds, but they do have access to use the NHS not just in an emergency capacity).
leave visa will set you back around £1000 and nationality the same.