Which of these two fits your description of a republic and which fits a democracy:
- Enforcing constitutional qualifications for candidates
- Ignoring the Constitution when a loud enough and large enough mob desires an unqualified candidate
This embodies a false dichotomy. The (small-"r") republican and (small-"d") democratic dimensions of the American polity cannot be tidily separated and treated as mutually exclusive elements that function independently of one another.
It is not a matter of "either / or", but of "both / and"
[1] - especially given that the republican
"enforcing [of] constitutional qualifications for candidates" is performed by democratic partisans (or their bureaucratic appointees/agents/payees/etc., who are themselves apt to be partisans and to act as such). As for the other prong of the dichotomy,
"loud enough and large enough mob" have never been required for "ignoring the Constitution". Indeed, the latter occurs far more frequently without the former than with it (hence, Michael Malice's observation that "'Our [republic-cum-]democracy' is always code for 'our hegemony'.").
[1] And as far as the "both / and" goes, the (small-"d") democratic dimension has waxed predominantly over the (small-"r") republican dimension. The republican dimension has waned significantly ever since the replacement of the Articles of Confederation with the Constitution, and on through Reconstruction (which gave us the 14th Amendment, the source of one of the bones presently being contended over), the 17th Amendment's implementation of the popular election of federal senators, etc. - to the point that America as a "republic" is little more than a collection of a few dozen or so pseudo-autonomous administrative districts of "our" federal democracy (LOL).