Are double-slit patterns really due to wave-like interference? According to various sources on the web, it seems like the general concensus is that there isn't actually any wave-particle duality with quantum particles. For example, this article implies that diffraction patterns in double-slit experiments were interpreted as wave interference due to apparatus limitation at the time they were first performed.
Does this mean that all those sources and animations showing two waves interfering are simply incorrect, classical conclusions which don't have anything to do with (quantum) reality?
What's actually the most confusing is that most sites which state that it is now possible to pass individual photons through these slits, also claim that these individual photons somehow interfere with themselves resulting in the observed patterns. That seems like a rather thin explanation, doesn't it?
So, is there actually any need to use wave interference to explain the phenomena, or can we simply state that the pattern is probabilistic in a certain way, without involving the "spooky" explanations? 
 A: Wave-particle duality is an old concept that doesn't have any meaningful explanation power. It's not how we approach quantum mechanics today. Truthfully, it's about as bad an idea to teach wave-particle duality, as it would be to introduce relativity with the detailed explanation of the ether, just to end that lesson with the phrase "and that's why the ether theory is all wrong". 
Yes, you are correct, classical waves are NOT a valid illustration of quantum mechanics, but they are a valid limit of quantum mechanics (for a large number of particles). This may sound like a contradiction, but it becomes clear, once you understand the difference between trying to find a classical explanation for quantum mechanics (which doesn't exist) and quantum mechanics being able to explain all of classical physics as a special case. 
It is also wrong to say that individual photons are interfering with each other. That's not what is happening, either.
You have a number of choices to approach this subject. You can stumble from one badly written webpage to another, or you can read Feynman's book about it: "QED: the strange theory of light and matter", which explains very nicely what quantum mechanics really is. Feynman does that much better than most of us, and I would give him the preference over anything that I, or most other folks could write about the subject.   
A: The main confusion comes from people not realizing that when one talks of wave particle duality the wave part belongs to the probability distribution  which can be calculated using the quantum mechanical solutions for the problem at hand. 
The solutions are called wave functions because they have sinusoidal expressions which are characteristic of the macroscopic wave. In contrast to water waves or pressure waves, the quantum mechanical wave description does not describe energy transport in space. The same experiment has to be done a statistically large number of times in order to accumulate a probability distribution for the double slits, for example. This one electron at a time double slit sequential exposures show this clearly.

A single electron is not a wave in space, its mass and energy running around wavelike. It just has a probability when it meets the boundary conditions of the two slits to scatter in a direction controlled by a wave probability distribution. In the beginning it looks random, the accumulation shows the interference pattern typical of wave equation. solutions.

So, is there actually any need to use wave interference to explain the phenomena, or can we simply state that the pattern is probabilistic in a certain way, without involving the "spooky" explanations? 

The "spooky" explanations come from people who do not understand the probabilistic nature of quantum mechanical solutions. Once one understands that, one can ignore spookiness. Sinusoidal functions appear in many solutions of equations called "wave equations" and interference patterns can appear in combinations. One has to be careful of the context where the solutions are used. In quantum mechanical frameworks we talk of probability distributions, as we talk of orbitals and not orbits in atoms, for example.
A: If you search this site for wave particle duality or something similar you'll find lots of questions addressing this and related issues.
The most complete description of particles we have is that they are excitations in a quantum field - this is called quantum field theory. Under some circumstances these excitations can behave like particles and under other circumstance they can behave like waves. If you take your example of the Young's slits experiment, it's possible to calculate the diffraction pattern using quantum field theory, but a quick glance at the paper I've linked should convince you that this is no easy matter. However in this experiment it's a very good approximation to use the wave model because the light is behaving very like a wave. And the wave calculation is simple enough to be taught to school children while quantum field theory is something you don't learn until postgraduate studies.
So while it might be technically true to say the diffraction pattern isn't being caused by waves, for all practical purposes we can treat it as if it is.
A: Even in the case of a single electron (as opposed to large number approximating a continuous wave), the probability distribution on the measurement shows double slit interference, rather than what would be expected in the case of a single slit.
So even though there is but a single electron, the physics of its journey is affected by both slits.  We have, through daily experience and classical education, the independent concepts of "particle" and "wave" behavior.  Neither fit exactly, but both are useful for explaining the quantum behavior of this single electron double slit experiment.

Does this mean that all those sources and animations showing two waves
  interfering are simply incorrect, classical conclusions which don't
  have anything to do with (quantum) reality?

No

What's actually the most confusing is that most sites which state that
  it is now possible to pass individual photons through these slits,
  also claim that these individual photons somehow interfere with
  themselves resulting in the observed patterns. That seems like a
  rather thin explanation, doesn't it?

It is an accurate explanation - one electron, two slits -> diffraction pattern prob. dist.

So, is there actually any need to use wave interference to explain the
  phenomena, or can we simply state that the pattern is probabilistic in
  a certain way, without involving the "spooky" explanations?

Words are arbitrary symbols, so you could describe the p.d. leaving out "wave" and using other language instead.  However, it seems to me that since the interference is explained beautifully by the double slit wave analogy, it is best to use it - just make clear that both particle and wave are analogies useful in describing quantum behavior.
