The purpose of the (US) government was and is to provide for the general welfare of its citizens; that's one of the two justifications given in the constitution for its ability to levy taxes.
Welfare is right there in the founding document, and a safety net is part of that.
Categorically, definitionally, and historically wrong.
What you are you are referring to is the following: “ Article I, Section 8, Clause 1: The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States…”
First, “welfare” here means “the state of well being.”
Second, it is tied to the “United States” as a whole - not any given individual, especially because the Supreme Court has ruled the government has no duty to protect its citizens from harm (Deshaney v. Winnebago County Department of Social Services, 1989; and The Town of Castle Rock v. Gonzales, 2005).
Thus, if the Supreme Court has established that the government has no obligation to protect citizens from harm, it has no obligations to economically provide for citizens either. That’s called “charity” and it’s what churches have typically done by collecting revenue (tithing) from its congregants.
20th century governmental usurpation of charity by rebranding it as “welfare” is a distinctly modern concept, that also happens to be constitutional, insofar that the government collecting taxes to distribute benefits on a needs basis does not violate the Constitution, which is entirely different than being enumerated in the document itself, which you erroneously conflated.
If such a constitution was written as you suggest, then it would have been outdated and torn up long ago. It would have been too rigid to stand the test of time. Also HN supports at least one emoji (囧).
There are several theories in designing constitutions. Many countries just give up and rely on common law instead. The more rigid constitutions tend to be ignored or the countries fall off into chaos, so America is somewhat successful in its implementation of constitutional law.
Chinese jiong should be the only emoji allowed on HN, since it is a valid Chinese character, but then so is a swastika (in both directions, non-nazi Buddhist meanings of course).
Welfare is right there in the founding document, and a safety net is part of that.