It kind of seems to me that the people wanting Charlie put down are doing so for their own convenience. They argue on one hand that since he's essentially brain dead, there's no point in trying to treat him, but then say that allowing treatment would prolong his suffering. What indications are there that he's suffering?

The parents should act as the decision-makers for their child, as children are not legally allowed to make most decisions for themselves. They want to allow further treatment, have respected doctors willing to do it, and have the financial means to provide for it. Why is this even a discussion?