I think I get what you mean by this. Quantum states violate classical logic (i.e., quantum systems can be in states that classically are mutually exclusive).
Here's the first issue: if we try to make quantum mechanics obey classical logic and insist that e.g., a system (such as an electron) travels through either one slit or the other, the theory fails completely. The "quantum maths" you referred to are what underlie the non-classical physics of quantum mechanics, the non-classical logic of quantum mechanics, and the "weirdness" that you seem to object to. In other words, if the "quantum maths" represent real systems then e.g., electrons must be in multiple (possibly infinitely) many places at once, they are never traveling along any trajectory, and they aren't even ever really in any point in space or spacetime ever. That's what the "quantum maths" say IF we interpret them as describing physical systems (i.e., if we adopt the many-worlds interpretation).
The second issue is that most interpretations (including the Copenhagen and orthodox interpretations) regard the quantum state as a probability function. That is, the wave function isn't a description of the state of a physical system but is a statistical entity yielding the likelihoods of measurement outcomes. It's true that according to such irreducibly statistical interpretations, the quantum maths don't describe objective systems, but they still describe an objective reality in that as in classical statistical mechanics quantum mechanics describes real, physical things, but these things are experimental outcomes/measurement results. Quantum states don't describe states of objective, physically realized systems they describe the probabilities that we will get particular objective, physically real results from particular experiments.
Finally, if we abandon the irreducibly statistical interpretations of QM, we don't somehow get a theory that follows classical logic or is anything like Newtonian physics. No-collapse interpretations like the relative state interpretations (the many-worlds interpretation is a relative state interpretation) don't result in any kind of classical reality. The MWI seeks to resolve the double slit experiment by stating that the electron goes through both slits and through neither, but in doing so it doesn't remain one electron (or rather, it remains one electron but now realized in different universes corresponding to possible outcomes). Even if one prefers this interpretation to the orthodox interpretation (or other interpretations), there remains a serious problem: by adopting this interpretation, we loose quantum mechanics.
Quantum mechanics is successful because it successfully tells us what the outcomes of experiments will be. Any non-quantum theory would entail the almost instantaneous destruction of every atom in the universe (among other disastrously wrong results). Quantum mechanics gives the right results. However, it is derived from the statistical structure of experiments: we can predict outcomes using the theory because we formulated the theory by repeatedly preparing and observing systems in the same way and determining the frequencies of particular outcomes. This worked. It is the foundation of quantum theory. A central issue with the many-worlds interpretation is that it holds that given any preparation of a system and subsequent measurement, there is no "collapse" because all possible results are realized. But the entire theory was constructed from and is rooted in the idea that we can predict the outcomes of experiments as follows: in the past, when we prepared X system and observed it in Y manner we obtained Z outcome with a particular frequency.
Let me simplify. One way of understanding that we have a 1/2 probability of getting heads when tossing a fair coin is by tossing a fair coin over and over again. As the number of tosses increases, the frequency of "heads" approaches 1/2. Of course, coins may not be fair: they could be biased or even be double-sided. The orthodox interpretation is like testing whether a coin is fair by tossing it many times and seeing whether heads occurs with a frequency approaching 1/2 (same with tails). The many-worlds interpretation is like taking a coin and assuming that it is a different coin every time we toss it, but determining that each one is fair if the frequency of outcomes of heads approaches 1/2.
There is no logical support for the statistical outcomes we actually get using quantum mechanics without assuming that the frequencies we get from repeating the same experiments only result in the outcomes we find. If we assume that all outcomes are realized, then no outcome can occur with frequently than any other. But in quantum mechanics, certain outcomes DO occur more often and are therefore more likely. That's what makes the theory successful.
The probabilities ARE the "wave's" amplitude. We don't treat the system as a wave. In fact, the whole "wave" approach is only one formulation of quantum mechanics and doesn't work in relativistic quantum physics. The "wave" component of quantum wave mechanics doesn't involve actually waves but uses wave amplitudes to derive probabilities. The first complete quantum mechanics was matrix mechanics and didn't involve any waves.
1) Newtonian mechanics requires that every electron in the universe plunge into its nucleus. It violates special and general relativity. It can't even deal with systems like electrons because it doesn't involve electromagnetic fields (or fields at all).
2) According to the orthodox interpretation, it is meaningless to ask what the photon or electron is doing when we aren't observing/measuring it. The results we get from measurement are due to the manner of measurements.
3) The MWI holds that the interference patterns emerge because all possible outcomes are realized, but can't explain why we get the patterns we do.
