This is misleading, because it gives the impression that the US back then had open borders in those pre-passport days. It did not, at least at Ellis Island and its counterparts at other port cities. Those disease checks were cursory because all such were cursory c. 1900.
More relevant, no one was allowed in, regardless of health check, unless they could prove that they had either financial resources to support themselves, or a US sponsor willing to provide such support. Ocean liner companies like Cunard, White Star, and Hamburg American prescreened their passengers, because those turned away were the transport companies' responsibility to carry back. (The same goes for airlines today.)