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It's quite clear: "These three qualifications excepted, the power to raise money is plenary, and indefinite; and the objects to which it may be appropriated are no less comprehensive, than the payment of the public debts and the providing for the common defense and `general welfare.' 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 should 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. [...] It is therefore of necessity left to the discretion of the national legislature, to pronounce, upon the objects, which concern the general welfare, and for which under that description, an appropriation of money is requisite and proper."

Hamilton's contention is indisputable and linked to his conception of the powers of sovereignty itself. No nation nor government will allow itself to face destruction if faced with such because there is a claim by some that its constitution would not allow action. The founders believed that there were implicit powers and there was little doubt about that, even Jefferson during the 1780s admitted as much.

A constitution is a fundamental law but the Nation is from a higher source since it creates that law. No constitution can restrict the government it creates to doing less than providing for the general welfare. That is what governments are created for in the first place.

Modern construction would have had "...throughout the United States:..." rather than "...;" had the list following meant that only the points enumerated were what was considered the "general welfare" Punctuation may have been different during those days but I have seen no convincing arguments that that was the case.