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In depictions of the double-slit experiment that model the photon or electron as a particle, i.e. when attempting to measure which slit the particle passes through, it always shows the particle entering one of the two slits. Why is it that the particle can't hit the space between or outside the slits, i.e. never even make it through? Is it implied that the experiment is just repeated until a particle makes it through, i.e. shows up on the film or detector on the other side?

I see how, modeled as a wave, the wave always makes it through. But the illustrations of particles kind of don't make sense to me. Is it because they are just simplified illustrations?

"Is it because they are just simplified illustrations?" you ask. The answer is simply: yes it is because they are simplified illustrations.

Furthermore, not only can the particle hit the barrier outside or between the slits, typically most of the particles do that. Only a small fraction make it through. I say 'typically' because in such experiments we don't normally bother to set up the optics (whether for photons or electrons) so as to restrict illumination to only the two slits and not the surrounding area. But in principle it could be done, and then only a few particles would miss the slits.

It is quite common, in experimental physics in this area, to do what is called 'post-selection'. That is the name for the practice of selecting from your dataset only those outcomes triggered by some signal, such as, in this case, the signal that a dot appeared somewhere on the final screen. Then after that the discussion is really saying not 'this is what happened in every run' but 'of those runs where something made it to the detector, this is what happened'. One can regard the simplified pictures as showing what is understood to have happened for those runs which were singled out by this 'post-selection'.

If you look up Doctor Quantum on youtube you'll find some (horribly dated) 3D animation videos that DO show the particles that bounce off the space outside the slits. At least initially in the 'marble' demonstration (the first light one also shows illumination on the slit device). They are culled later because of the same reason they've been culled from other examples of the double slit experiment.

And that reason these particles aren't shown is because they are irrelevant: they don't pass through the slits and so are not part of the "what happens when they pass through the slits" experiment.

Its kind of like asking why people under the age of 18 aren't included in election polling data. Surely these people exist!

When I performed this experiment the last time, I used a laser, so no single photons were fired, but an endless stream of photons, so to say. Then, the double slit was so positioned that the maximal intensity way measured at the detection screen (with a photometer).

That some photons hit the area between or outside the slits is very likely, as the laser itself has a certain cross-section. Then, the wave-function would collapse or the photons are reflected and therefore not measured.

In theory, if you perform this experiment with one photon, what would you expect? Well, your wave function should include the possibility of a reflected (or not visible / detected) photon with a non-zero probability. Then, an experiment on it would only make sense with many many measurements, to have a reliable statistic for you probability-distribution, so you end up with doing the experiment many times.

Here is a double slit experiment one electron at a time hitting the screen:

The quantum mechanical setup is : "electron of given momentum scattering on two slits a fixed distance apart , a fixed width". The screen shows the electrons which go through.

The quantum mechanical solution, i.e. the wavefunction $Ψ$ whose $Ψ^*Ψ$ gives the final probability distribution shown in the last frame, gives the probability for the electron to go through the "barrier of two slits" and hit the screen.

Though the distance between slits is small, if one could place a detector there, or a photo-luminescent material, there will be a probability of detecting the electrons having a probability to hit there. One does not need to confirm this, as the wave nature of the probability is evident in the interference pattern.

But the illustrations of particles kind of don't make sense to me. Is it because they are just simplified illustrations?

It should be clear from the above then, that the electrons are not classical particles, and should not be extrapolated on a two dimensional graph as particles, except as an approximation over large distances where they do not interact.