Oh, I must disagree. The US Constitution is very much concerned about the interaction of citizens and the federal government - not just states and the federal government.
The purpose of the 14th Amendment was to extend the protections of the 5th Amendment to the states, which heretofore, had only been between the Federal government and its citizens. Doesn't that imply that in fact the Constitution recognized that it would interact directly with the citizens. The protections were intended to regulate this interaction in favor of the citizen, not to eviscerate it completely.
The creation of a Court system and its deliniation of jurisdiciton presupposes a measure of control by the Federal government of its citizens, especially in the interstices of state actions (perhaps through the exercise of an enumerated power, the necessary and proper clause, or the tax/spend power). While the Constitution does prescribe a very large role for the states in governance, it is in fact a document of control (e.g., the existence of separate federal crimes).
The STATED purpose of the Constitution was to LIMIT the Federal Government by only enumerating its specific powers. By claiming "General Welfare" applies to anything and everything, you have just removed any limit at all. There is no need for amendments, one need just proclaimed it falls under "General Welfare" and it is now covered.
The term does NOT mean what you state it means.