No less of an authority than Albert Einstein was always uncomfortable with the whole wave function/probability distribution/fundamental uncertainty "thing" which is central to quantum mechanics. As he once famously said, "God does not play dice". However, the indeterminacy of quantum mechanics comes straight from the basic mathematics of the theory, and as Niels Bohr famously replied to Einstein, "God not only plays dice, he throws them where they cannot be seen".
The theory of quantum mechanics can be reformatted, so to speak, such that the electron does have a specific, but unknown position, as opposed to being everywhere at once with differing probabilities. Such theories are called hidden variable theories, for the obvious reason. In most situations standard quantum mechanics and the hidden variable theories predict exactly the same thing, so there isn't much practical difference between the two. However, there are complicated, somewhat exotic situations in which they do not predict the same outcome, and physicists have examined these. The experiments are challenging to carry out, and one or two have in fact seemed to favor hidden variables over quantum mechanics. However, the great bulk of the experiments favor standard quantum mechanics, and as time goes by quantum mechanics is lengthening its lead over hidden variables. It would seem that the Universe is perfectly happy to be a place where "particles", at the most smallest and most fundamental level, simply
do not exist.