If the person is able to tell you that they have no insurance, this by itself means they're able to talk reasonably, this means they are conscious and can breathe and so (absent heavy ongoing bleeding e.g. from a gunshot wound) they aren't dying right now - so urgent emergency healthcare isn't really required.

If they can move on their own, then they can get to the clinic of their choice themselves if and when they choose to; if they can't (which can be caused by anything from a sprained ankle to a fractured hip) then they still may prefer to be delivered to the hospital by family&friends or a taxi instead of paying thousands for an ambulance delivery.

good point about using that as an assessment of the persons state of being. they still could be bleeding however which needs to be stopped unless it's just surface scratches, or have broken bones that need temporary stabilizing.

it seems other than for the first point the question only becomes relevant after any relevant first-aid actions are completed.

i have been in a situation where a guy fell out of his wheelchair. he wasn't injured, he just could not get back into his wheelchair and he was to heavy for any of us to help him without us risking to drop him in the process which might have then actually risked an injury. he looked like the kind of person who might not have had insurance. though he didn't make any statement about that and no-one even considered that to be an issue. we saw no choice but to call 911, which sent an ambulance with people who could help (and they knew him too).

i wonder now how we would have reacted if he had told us that he has no insurance. we might have been looking for alternative help with an outcome that could have been good or bad.

with that in mind i'd think other than for the reasons you mentioned the statement should be ignored.