The problem with that is that emergency care is absurdly expensive. I was in an accident last year that put me in hospital for one night for observation, and I was billed almost $20k. Where illness rather than accident is the problem, people postpone going to a doctor or hospital until they are really ill, even though it would be much cheaper to address a minor illness
before it becomes a medical emergency - a stitch in time saves nine, and all that. But people without insurance find access to non-emergency care difficult because of cost, and so the emergency spend half their resources dealing with completely avoidable medical problems.
And even if you do have insurance or the ability to pay, or (AFAIK) if a hospital waives payment for someone indigent, there's still a metric ton of paperwork to be filled out, which makes doing my taxes seem simple and fun by comparison. Don't even get me started on the catastrophically bad state of medical record-keeping.