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.
