Of course Hamilton said a little more than your out of context little snippet of less than a sentence Straw Man. I indicated in red the context.
The Madison argument, despite its frequent use by hateists, fails the I-didn't-just-fall-off-the-turnip-truck test because it defies the Constitution as determined by the SCOTUS. The Hamiltonian interpretation was settled by the SCOTUS long before FDR!!!
Hamilton and Jefferson fought about the meaning of "general welfare" in several important cases:
"Opinion on the Constitutionality of the National Bank," to establish a national bank to handle taxes, borrowing and debt payments.
"Report on Manufactures,'' seeking government support for industry to keep the economy strong and well-supplied in case of war.
John Marshall, Chief Justice of the United States from 1801 to 1835, decided that the Hamiltonian view was established as our fundamental law. Marshall's 1819 opinion in the case involving the National Bank, (McCulloch v. Maryland,) is a milestone for the confirmation of the national government's exercise of its implied powers
Go ahead and look at the entire sentence. What he's saying essentially is that Congress is authorized
"in express terms" to appropriate money, but that they can't use it for anything that's not authorized by the Constitution,
"either expressly or by fair implication".
Hamilton was NOT going around suggesting that Congress could act in arbitrary ways, counter to the written Constitution. He was a bit of an asshole, often defaming his contemporaries and
out ******* someone else's wife at the time of
Report on Manufacturers 
.... but even he would not have gone so far. In fact, in
Federalist 84, he writes:
"I go further, and affirm that bills of rights, in the sense and to the extent in which they are contended for, are not only unnecessary in the proposed Constitution, but would even be dangerous. They would contain various exceptions to powers not granted; and, on this very account, would afford a colorable pretext to claim more than were granted. For why declare that things shall not be done which there is no power to do? Why, for instance, should it be said that the liberty of the press shall not be restrained, when no power is given by which restrictions may be imposed? I will not contend that such a provision would confer a regulating power; but it is evident that it would furnish, to men disposed to usurp, a plausible pretense for claiming that power."
And also:
"There remains but one other view of this matter to conclude the point. The truth is, after all the declamations we have heard, that the Constitution is itself, in every rational sense, and to every useful purpose, A BILL OF RIGHTS. The several bills of rights in Great Britain form its Constitution, and conversely the constitution of each State is its bill of rights. And the proposed Constitution, if adopted, will be the bill of rights of the Union. Is it one object of a bill of rights to declare and specify the political privileges of the citizens in the structure and administration of the government? This is done in the most ample and precise manner in the plan of the convention; comprehending various precautions for the public security, which are not to be found in any of the State constitutions."
So firm is Hamilton in his assertion that the U.S. Constitution LIMITS the power of federal government that he doesn't even believe a
Bill of Rights is necessary!
Hamilton's
Report on Manufacturers has been repeatedly used as a feeble excuse for federal power grabs. Almost everything written during the era runs counter to the twisted interpretation used by statists, even Hamilton himself.
Further, common sense tells us that if anybody had been running around claiming that Congress could make arbitrary law on whatever interpretation of
"the General Welfare it took a notion to make... the Constitution would NOT have been ratified! The 10th was included in order to put to rest any unease about this.
The proof that previous, statist usage of Hamilton is a depredation upon the original meaning of the document and a HUGE MISTAKE is readily witnessed in the fact that
the entitlement programs brought to life by it are breaking us. 
IOW, it's lead us so far astray that we now face an existential threat to the country itself through our debt load.
It was wrong then... it's wrong now. But rather than correct a deviation from the true path laid out for us by our founders, leftists would like to BUILD on that mistake in order to further deprive us of the freedom that was our birthright, and all so they can get some free cheese from the government.
McCulloch did not give Congress the authority to make up any "implied powers" it might want. It gave an opinion on the
Necessary and Proper Clause which allowed "implied power" for the purpose of carrying out it's ENUMERATED powers under the U.S. Constitution. In that decision, Marshall wrote:
"We admit, as all must admit, that the powers of the Government are limited, and that its limits are not to be transcended. But we think the sound construction of the Constitution must allow to the national legislature that discretion with respect to the means by which the powers it confers are to be carried into execution which will enable that body to perform the high duties assigned to it in the manner most beneficial to the people. Let the end be legitimate, let it be within the scope of the Constitution, and all means which are appropriate, which are plainly adapted to that end, which are not prohibited, but consistent with the letter and spirit of the Constitution, are constitutional"
Bottom line... There's no enumerated power which allows Congress to manage our healthcare or to interpret whatever meaning it might want from the phrase "General Welfare". Note, that only the most retarded among the idiots on Capitol Hill are bothering with "the General Welfare" argument. Most are claiming authority through "the Commerce Clause"... which is also a FAIL because inactivity is not commerce, even by the modernized definition of the word.