What would be the best way to understand the relation between amplitudes and probabilities in the “non-quantum” world?
In your NYT piece you say that it would be weird to give a “sq(-1) probability of rain tomorrow”, but are there relatable uses or cases for amplitudes outside of quantum physics?
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It's not obvious why you'd want to talk about complex amplitudes, whose squared absolute values are probabilities, outside the context of quantum mechanics. Maybe a better way to say it is: once you're talking about such amplitudes, essentially by definition you ARE talking about quantum mechanics. :-)
Having said that, in classical probability theory, it's sometimes useful to look at the (positive) square roots of probabilities, for example to get a distance measure between probability distributions. See here:
https://en.wikipedia.org/wiki/Bhattacharyya_distance
Also, we've often been able to use quantum tools to prove new results even about classical theoretical computer science. See here for a beautiful survey, though one that's already a decade out of date:
https://arxiv.org/abs/0910.3376
In this way, the mathematical tools that we've developed in quantum information can "pay rent" in classical CS, even if we counterfactually imagined that our world wasn't quantum-mechanical at all. It would take some time to go through an example of this, but (e.g.) it was shown that if a certain classical error-correcting code existed (called a 2-query locally decodable code of subexponential size), then an even better quantum error-correcting code would also exist, but the latter was something that people already knew how to rule out.
In these situations, some people would argue that we're not "really" using quantum physics or amplitudes; we're just taking MATH that was developed for that purpose, and repurposing it for something else. But it would be weird to say in a talk, "I'm now going to introduce a unit vector of complex numbers that's just an ordinary vector, nothing physical at all about it, then apply the following tensor product of 2x2 and 4x4 unitary matrices to it...," when everyone knows full well that all your intuition about this came from quantum states. Indeed, I confess that even when I'm doing linear algebra that has nothing to do with QM---i.e., my vectors really are just vectors of real or complex numbers, not amplitudes---I sometimes slip up and use the physicists' notation for quantum states (called the Dirac ket notation).