Electron vs electron density
Of course, electrons do exist. We have a myriads of observations confirming their existence. What does not exist is working classical picture of an electron. Electrons can behave in certain circumstances like waves, in some other - like particles. That is already nonsense from the classical physics point of view with its universal dichotomy: something is either a particle or a wave. 

But the trickiest part of the wave-particle duality is that despite the fact that electrons can behave as waves and as particles, their are, strictly speaking, neither of them. Our language developed on the basis of observations of macroscopic surroundings with its well-established classical notions of waves and particles is simply inappropriate for the description of the microscopic phenomena. Thus, whenever one tries to use any language except for that of mathematics to describe microscopic world he/she quickly runs into apparent paradoxes.

With respect to the desire of pictorial classical description of an electron I feel like the only thing I have to do is to quote Paul Dirac:


  […] the main object of physical science is not the provision of
  pictures, but is the formulation of laws governing phenomena and the
  application of these laws to the discovery of new phenomena. If a
  picture exists, so much the better; but whether a picture exists or
  not is a matter of only secondary importance. In the case of atomic
  phenomena no picture can be expected to exist in the usual sense of
  the word 'picture', by which is meant a model functioning essentially
  on classical lines. One may, however, extend the meaning of the word
  'picture' to include any way of looking at the fundamental laws which
  makes their self-consistency obvious. With this extension, one may
  gradually acquire a picture of atomic phenomena by becoming familiar
  with the laws of the quantum theory.