I don't think it's inconsistent. The people who advocate for free healthcare would probably also advocate for other parts of a social safety net, like food banks, food stamp programs, free access to water, temporary housing for the homeless and programs to get them to permanent homes, etc.

And indeed the US has many such services, though they are often overwhelmed, or performed through a complex set of nonprofits combining govt funds with donor money as best they can to provide good support. Sometimes they are overwhelmed specifically because of externalities related to the for-profit insurance healthcare system. The high and often unpredictable cost of any medical situation, means people with tight budgets become sicker before getting care, if they ever get it, leading to a higher percentage of bad outcomes, up to and including job loss and homelessness for the individual, and knock-on effects for the rest of their family. This puts extra pressure on the parts of the support system that do exist - those already mentioned, and, of course, the police, who end up getting called to many situations that could have been prevented by the people involved having better services to begin with, to meet their health, food, and housing needs.