“Ascribing a charge” is not quantizing it except as a mere example for an explanation of some relationship. It is giving an estimate of value to something that can never be perfectly precisely measured. They know that the decimal representation of a value is limited to the number of digits that can be recorded and accurately measured. Classical physics is not proposing that a given number is perfectly exact. Although they do propose that relationships between values are perfectly exact (calculus, which wouldn’t work in a quantized universe. But it does work in the real universe).
Right off the bat, electrons don’t “emit” charge. They are the result of charge, a center of concentration of the charge field.
There is a problem with that (besides the above mentioned issue).
Each electron would emit only one wave in front of itself (given that it “emits”). It could possibly interact with its own emission, but only once. That would cause that particular electron to shift its course.
The next electron has no idea where the first electron went. Its particular path is slightly different than the first. So when it has the same kind of interaction with its own emitted wave, it takes a still different course.
The question is; since the electrons seem to gather in certain places and avoid others, how do the electrons know where on the screen to gather and where to avoid?
According to Quantum Physics, they consider if they are being watched and choose based on that.