Philosophers (Popper in particular) have long debated the interpretation of probability as applied to the world. Their concerns echo those of physics (frequency, preparation, ensembles, ). It would seem that probability is the root of the issues, not QM per se.
The meaning of probability is an issue in QM because it doesn't appear to represent states of ignorance/knowledge as it does in the classical interpretation of probability, but is interpreted ontologically.
The other related issue is that of probability amplitudes. An amplitude is squared to get a probability. Amplitudes also interfere with each other. This is at the root of the super-position of states and entanglement. Ontologically this is difficult to interpret classically. The former knocks up against determinism, and the latter against locality.
I can't resist pointing out here that mathematicians have already been here before. The concept of the imaginary, as the square root of -1 was struggled with over centuries. Today, it's been settled and students are introduced to it routinely at school.
Here we have the analogous phenomenon in physics, the amplitude which holds the secret and mystery of QM is the square root of probability.
This too has a precursor. In antiquity, the square root of two was discovered not to be a rational number, but irrational. This had to wait until the modern era for the notion of the real line continuum when this was put into context. Today, we just think of the square root of two as just a point on the real number line.
It seems every era has its trouble with square roots...!
Quantum mechanics implies a formulation in which from initial data a (classical) probability distribution is derived, so that (unless we have a single possibility, with 100% probability) "facts" cannot be deduced, we only infer propensities.
Experimental scientists have kept applying classical probability theory, the world has always been stochastic. What is special of quantum mechanics - and behind the measurement problem - is the existence of a continuum of alternative, incompatible sample spaces. In the simplest system, the qubit, there's a different sample space - corresponding to a different observable - for each direction in (euclidean) 3D space; only one of these observables can be measured at a time, when the other observables do not make sense (are not defined physical properties). The qubit has a continuum of states that appear in pairs, while the classical bit has only two states.