Not all waves can be split, e.g. Solitons which are named because of their particle like nature and can be observed in optics, water chains of oscillators and more.
I think that is the crux of the issue, we have waves with a discrete energy, we can call them particles but they are very different from the traditional image that people have of what a particle is.
You can easily split solitons in water, just cut it in the middle and both sides will continue to live on as the water displacement field is still there.
On the other hand, try grab a part of an electron cloud around an atom and you will either get the entire electron or you will grab nothing. No matter what you do you can't separate one part of the cloud from the other, they are always connected. Trying to grab the electron will either remove all of the wave parts outside, or remove all of the wave part you try to grab. There is no classical system that behaves like this.
I think that is the crux of the issue, we have waves with a discrete energy, we can call them particles but they are very different from the traditional image that people have of what a particle is.