I don't believe to be an unreasonable conflation. Yes, you're correct that taxpayers cover all uninsured insurance payments, but car accidents constitute a larger percentage of events versus your example of suicides. Also, car accidents are not always the fault of the driver (hence the term "accident")... so they are not only in the victim's control... hence we need to be defensive about it. It's like wearing helmets and pads in a football game... players don't only hurt themselves, but carry a higher threat of being hurt by others... hence the pads and such. If they stopped wearing such equipment, would you be quick to say the taxpayers should cover their injuries too?
On your second point, yes, I agree that suicides shouldn't be covered by taxpayers either. The problem is, you can't know it was a suicide until AFTER the emergency has passed... so you simply can't segregate the incoming patients to the ER. But since they aren't as frequent/common as car accidents, I think it's something a civilized society should just accept and deal with. Hell, that's all the more reason to support suicide prevention and such, but that's a totally different point.
As for the healthcare topic, I agree with you there - I, too, would support this system as you would. But I still think there are limits to what's considered acceptable. In your example of the mortgage payment, would you be ok with paying the mortgage, and then having the resident damage the place to the point of worthlessness? Does that seem fair to you? Unless you want to pay the mortgage with no expectation of any return value of any kind... in which case you're just talking about "social charity" essentially... which is admirable, but not really relevant to the topics here.