This is more a comment than an answer, but it's too long to put it in a comment.
If you do the double slit experiment with a laser beam, you get an interference pattern...
That is an unnecessary restriction. Even with sunlight one get an intensity distribution behind a double slit:
I made a small hole in a window-shutter, and covered it with a piece of thick paper, which I perforated with a fine needle... I brought into the sunbeam a slip of card, about one-thirtieth of an inch in breadth (about 0.85 mm), and observed its shadow, either on the wall, or on other cards held at different distances. Besides the fringes of colours on each side of the shadow, the shadow itself was divided by similar parallel fringes, of smaller dimensions... Thomas Young 1803
Even with incoherent light, Young achieves a result by reducing the incident light to a point-like source. This makes sense because a wider beam works like many pinholes. Each hole creates its own intensity distribution behind slits and there are no fringes visible. A further explanation of the pinhole highlights that the pinhole makes incoherent light coherent. But how a pinhole makes light from a thermic source - for example from a sodium-vapor lamp - coherent is not discussed. Furthermore, behind a pinhole the light has to have the same properties as laser light.
There is a point in Young’s explanation, that seems to be important. Young did not realize, that the image with water waves, he used to explain the fringes as interferences, has a weakness:
In point A and B waves are produced with their crests and troughs and by this the points C, D, E and F are not stationary, they are moving. The sketch is only one moment of what you will see in a pond. But in double slit experiments with light the intensity distribution was observed as a stationary pattern. Didn’t this phenomenon needs further investigation and explanation?
... as light goes through different paths and the phase shift leads to constructive and destructive interferences. This gives us an evidence of wavy behavior of light.
It is always good to break down experiments to pieces. To produce fringes it is not necessary to use a double slit. It’s not necessary even to use a single slit or pinhole, like an Airy disk. Behind every sharp edge an intensity distribution is observable. In fact, for the deflection of electrons first was set up an experiment with a thin wire. So the explanation of fringes has to contain the simplest case of edge diffraction. Than perhaps it is not necessary to refer to interferences from different slits?
Now, if you do the same experiment with one photon (or electron ,...) at a time, you get the same interference pattern, but this time the interference pattern gives you the probability of each photon ending up in a specific place on the backstop.
“Single” photon experiments are the next step to break down the slit diffraction to pieces. It is explained that the particles - if not disturbed by measurements-interact with itself and thus diffracts. But remember, that any measurement detects the particle as a whole. Even in any close distance to the slits. Sometimes from the left slit and sometimes from the right slit. To prove this it is enough to put the observation instrument close to the slits and fire single photons.
My confusion is, that why don't we interpret shooting with a laser beam just like shooting photons one at a time? If we think of a laser beam as millions of photons, that we are shooting them at the same time, why can't we just think of each photon as following the wave function (which is a probability function) and ending up on a spot on the backstop?
You can think like that. And you get the same result as from a statistical point of view. The question is, did you get more information about the process?
Why we think of the laser beam as a real wave that interacts with itself to make an interference pattern?
This is a really interesting question. A wave has the property of swelling intensity. Lasers could be used in a Continuous wave operation mode, where the output power is constant over time. Such a beam doesn’t has the behavior of a wave. The photons of a light beam are oscillating with their electric and magnetic field components and by this could be seen as waves, moving through empty space.