Without money, how can a hospital afford to provide health care? There is no such thing as free health care. Someone has to bear the expense of its provision in some form. Hence a hospital has to worry about what you consider to be its second task at least as much as its first task.
I understand that, but I don't think it's really 'right' to be pestering people about debt before they are discharged. Afterwards, as much as you want. But then, I don't work in the medical field -- one of the beauties of privatization is that if that's how you want to govern your business, by all means you can do that.