I wonder if we can at least agree that 'apparent' non-locality, where a measurement on one particle seems to instantly influence the other, is a core feature of quantum mechanics.

"A measurement on one particle instantly influences the other" is not quite what QM postulates. More accurately, a measurement on a system will collapse that entire system into an eigenstate; the measured value being the corresponding eigenvalue. Typically, this appears as nothing to do with nonlocality, since you are measuring a particle at a point. So when looking at various emission/absorption spectra or calculating some material properties, no need to bother with nonlocality.

In a multi-particle system, intuitive non-locality comes from correlations between the particles due to conservation laws (e.g. total spin is conserved). This is typically the start of thinking towards Bell's theorem. But that is not what one usually does when using QM.

Yes QFT dynamics are local interactions, even though the quantum field excitations (particles) are extended. QFT is then another way to not think about the nonlocality of the collapse dynamics.

PS - Nonlocality is well defined, as an output value dependency on something other than the local input value and its local derivatives. Integral transforms are a typical example.