There shouldn’t be a distinction between quantum and non-quantum objects. That’s the mystery. Why can’t large objects exhibit quantum properties?
What makes quantum mechanics distinct from classical mechanics is the fact that not only are there interference effects, but statistically correlated systems (i.e. “entangled”) can seem to interfere with one another in a way that cannot be explained classically, at least not without superluminal communication, or introducing something else strange like the existence of negative probabilities.
If it wasn’t for these kinds of interference effects, then we could just chalk up quantum randomness to classical randomness, i.e. it would just be the same as any old form of statistical mechanics. The randomness itself isn’t really that much of a defining feature of quantum mechanics.
The reason I say all this is because we actually do know why there is a distinction between quantum and non-quantum objects and why large objects do not exhibit quantum properties. It is a mixture of two factors. First, larger systems like big molecules have smaller wavelengths, so interference with other molecules becomes harder and harder to detect. Second, there is decoherence. Even small particles, if they interact with a ton of other particles and you average over these interactions, you will find that the interference terms (the “coherences” in the density matrix) converge to zero, i.e. when you inject noise into a system its average behavior converges to a classical probability distribution.
Hence, we already know why there is a seeming “transition” from quantum to classical. This doesn’t get rid of the fact that it is still statistical in nature, it doesn’t give you a reason as to why a particle that has a 50% chance of being over there and a 50% chance of being over here, that when you measure it and find it is over here, that it wasn’t over there. Decoherence doesn’t tell you why you actually get the results you do from a measurement, it’s still fundamentally random (which bothers people for some reason?).
But it is well-understood how quantum probabilities converge to classical probabilities. There have even been studies that have reversed the process of decoherence.