True, healthcare isn't mentioned specifically, but Hamilton's view of general welfare was expansive and that it encompassed anything that bettered the public wellbeing. I think it's pretty hard argument to make that healthcare doesn't promote the public wellbeing. Just as it's difficult to argue that providing for the common defense is limited to providing only the means of defense that were available at the time the constitution was written.
Anyway, the Supreme Court agrees with my "distorted thinking" and my "personal desires" on this issue, so it is the law of the land.
Here is Hamilton in his own words.
The terms "general Welfare" were doubtless intended to signify more than was expressed or imported in those which Preceded; otherwise numerous exigencies incident to the affairs of a Nation would have been left without a provision. The phrase is as comprehensive as any that could have been used; because it was not fit that the constitutional authority of the Union, to appropriate its revenues shou'd have been restricted within narrower limits than the "General Welfare" and because this necessarily embraces a vast variety of particulars, which are susceptible neither of specification nor of definition.
The only qualification of the generallity of the Phrase in question, which seems to be admissible, is this--That the object to which an appropriation of money is to be made be General and not local; its operation extending in fact, or by possibility, throughout the Union, and not being confined to a particular spot.