The measurement, i.e. the Born rule, is just as fundamental as the wavefunction. The wavefunction doesn't mean anything on its own, it's not a measurable quantity that can be used to make any observable prediction whatsoever. If I claim that the wavefunction amplitude for some electron being at some location at some point in time is 1/2(1+i), how would you verify this prediction without invoking the Born rule?
You may say that nuclear fusion in stars does not mean anything; only what matters is that we see light. At a certain level it is true, but to simulate a system we need to simulate its inner workings, not only - the end effect.
The exact phase of a wavefunction does not matter - but it is an important phenomenon, giving raise to gauge invariance. The Born rule can be derived. In short, since we use unitary operators, length is preserved. For a derivation, see https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.05....
Also, to be nitpicky, we also never measure probabilities. Something (macroscopic) happened or not. It gives rise to quite fundamental and philosophical questions, including "what is (classical) probability" (I don't know an answer that fully satisfies me), many world interpretations (maybe all possible things just are), and in general what on indeterminism and free will.
I'm not a quantum physicist and so can't really comment on why, but it's clear that the paper you linked is not widely accepted, as the Born rule is still taught as a postulate of quantum mechanics, not a derived property of the wavefunction. I'd wager a guess that the paper ends up inventing some other postulate that is itself not derivable from the wavefunction, so it becomes at best a philosophical matter which postulate you actually prefer.
I also don't agree with your comparison of what I said to the nuclear reactions happening inside a star. The problem with the wavefunction without the Born rule is not that it's difficult to observe, it's that it's literally meaningless: knowing the value of the wavefunction for some state of a system doesn't tell you anything at all unless you apply the Born rule to this value.
And as for probabilities, certain kinds of probabilities at least have a very clear and simple definition (though they are rather narrow cases): if you repeat an experiment in exactly the same conditions N times, and an outcome O happens in p/N times and doesn't happen (1-p/N) times, then we define P(O), the probability of outcome O, as the value p/N. For systems where this applies, it is very much a measurable quantity (with some noise, of course, related to the fidelity with which you can reproduce the same experiment).
I do agree that this well-defined, measurable, concept of probability is rarely what we mean by "the probability of O", since (a) it's often hard or impossible to repeat (or even perform) the experiment, and (b) we often care about what will happen the next M times we repeat this experiment, and the measure P(O) I defined above does not tell us anything about future events.
Wavefunction has the Born rule, it's just not an independent postulate, but an emergent behavior from the Schrodinger equation. Also knowing the wavefunction does tell you everything, all properties are derived from it.
You say you need the Born rule to understand what's going on, for this you don't need it as a fundamental phenomenon, you only need to eventually observe the Born statistics, which is sufficient to provide understanding for you.
Again, the wavefunction just tells you "the amplitude of the state in which the photon is at location (x, y) on the screen at time t is sqrt(i); the amplitude of the state in which the particle is at position (x+2,y) at time t is sqrt(1+i/2)". Given these numbers, where do you expect to find the particle at time t?
Sure, but how specifically do you think it was checked?
Actually, I'll tell you how it was checked: they ran lots of experiments, and confirmed that the probability to find the particle in one state or the other is precisely equal to the norm of the wavefunction of the respective state. Also known as the Born rule.
So the amplitudes have no physical meaning directly, it's just that their norm represents the probability of the state being observed. That is, you have to take the Born rule as an additional postulate that is entirely separate from the wavefunction.
Now, you can dress this in other language. Some versions of MWI say that the universe splits into many literal worlds after any quantum event, and the number of worlds in which it has a certain outcome is proportional to the norm of the amplitude of the wavefunction of that outcome; based on this, they then derive the Born rule as P(stateA) = num_worlds(stateA) / num_total_worlds = norm(|stateA>). Of course, this is still the Born rule, and it is still not derivable from the wavefunction, still an additional postulate - just with extra steps.
And I don't know what you mean when you say that the Born rule is not statistics: it is exactly statistics (or at least probabilities, if you make a distinction). Sure it's possible to get a million tails in a row, that is always possible in statistics - by definition, any event with probability higher than 0 is possible.
What's observed is statistics, not the rule, and the goal of modelling is statistics. Once you have statistics, you don't need to assume the rule, because statistics tells you what you want to know - quantitative properties of the process. Also since statistics is quantitative, it can be just computed without interpretation, such quantitative properties don't depend on interpretation, simply because they are computable. Maybe you're confused by assumption that rule is identical to statistics, and thus believe statistics uncomputable merely because the Born rule is uncomputable? The fact is the Born rule allows to miscalculate statistics, because the probabilities are unintuitive, there is a precedent.
Amplitudes as quantitative properties are sufficient for calculation of statistics. Ironically classical theory of probabilities works the same way: first it assigns arbitrary weight to outcomes, then divides them by the weight of ensemble (usually >1 contrary to QM) to get statistical coefficients. The weights can be scaled by any constant factor, and the calculation still works.
It did that by checking whether an electron is found at a particular position relative to the nucleus lots and lots and lots of times, and building a heatmap of where the electron was actually found in ach individual experiment; the heatmap of course corresponded to the wavefunction model. So the experiment found that the probability of finding the electron at a certain position exactly corresponds to the square of the amplitude of the wavefunction at that position, i.e. the Born rule.
What the experiment did NOT do is directly detect the wavefunction of the electron, because that is, again, not a phsycially meaningful quantity.
