Why do we need the third axiom of QM to explain the wave function collapse? Why don't we use the decoherence process as an axiom? I've always seen the standard interpretation and axioms of QM as in some way tricky on a philosophical level. They state the collapse of the wave function is caused by the measurement.

3.b If $A$ is an observable with eigenvalues $a_k$ and eigenvectors $|k\rangle$ $(A|k\rangle = ak|k\rangle$), given a system in the state $|\psi\rangle$, the probability of obtaining $a_k$ as the outcome of the measurement of $A$ is $p(a_k) = |\langle k | \psi \rangle|^2$. After the measurement the system is left in the state projected on the subspace of the eigenvalue $a_k$ (wave function collapse)
source: https://ocw.mit.edu/courses/nuclear-engineering/22-51-quantum-theory-of-radiation-interactions-fall-2012/lecture-notes/MIT22_51F12_Ch3.pdf, page 15

I've known the general consensus is that the collapse of the wave function is a manifestation of a more general process called decoherence. So in the standard interpretation, the wave function seems to collapse due to the environment, but it really collapses due to the measurement. Maybe because in the first experiments of the physicists of the 20th century the measurement apparatus caused the collapse of the wave function, so it might have seemed the best way to define those axioms..
Why don't we use the decoherence process as an axiom somehow?
In this way, we might avoid some misinterpretation of quantum mechanics and have a deeper understanding of the subject.
 A: Many people have many different ways of stating the axioms of quantum mechanics, including some formulations that look nothing like the typical presentation in a freshman physics textbook. Not worrying too much about completeness and perfect rigor, a pretty standard way of formulating such a set of axioms would be:


*

*Wavefunctions exist in a Hilbert space.

*The time evolution of the wavefunction is unitary.
There is nothing about Copenhagen or wavefunction collapse, and nothing like that is needed. If you want wavefunction collapse, you're free to add that as an axiom. For a treatment in this style, including optional Copenhagen stuff, see Carroll and Sebens, "Many Worlds, the Born Rule, and Self-Locating Uncertainty," https://arxiv.org/abs/1405.7907 .
Decoherence is not an additional hypothesis to be added on to the standard axioms of quantum mechanics. Decoherence simply results from the axioms. For comparison, the standard axioms of arithmetic imply that 2+2=4. You could add an axiom saying 2+2=4, and the system would still be consistent, but the additional axiom would be superfluous.
An interesting paper on this type of thing is Allahverdyan, Balian, and Nieuwenhuizen, "A sub-ensemble theory of ideal quantum measurement processes," 2017, https://arxiv.org/abs/1303.7257 . They make a toy model in which you can see decoherence and pick out processes that look very much like the wavefunction collapse of the Copenhagen interpretation. One thing that comes out naturally is that there are various time scales involved. Since the Copenhagen interpretation is normally described in terms of instantaneous collapse, one possible point of view is that Copenhagen is just an approximation, which can fail to hold in the real world.
The most austere versions of the Many-Worlds interpretation ("MWI-lite") are immune to this kind of attack, since they only posit postulates 1 and 2 above, which everyone agrees on anyway. (This is the point of view advocated by Carroll and Sebens.) In this point of view, we never even talk about things like "worlds" or "branching" of worlds. If, on the other hand, you want a more baroque version of MWI ("MWI-heavy") in which we talk about these things, then MWI-heavy is probably only an approximation to the exact results of standard quantum mechanics, for the same reasons that CI can't be expected to be exact.
A: 
So my question is: why don't we use the decoherence process as an axiom somehow?

The third axiom you quote is necessary in relating the mathematics to real numbers measurable in the laboratory. It is responsible for describing the atomic spectra, one of the main reasons quantum mechanics had to be invented.
Decoherence might be shaped into an axiom, with too much mathematics, not a one to one relation to observables. It is a "theorem", and as in pure mathematics a theorem can become an axiom and the axiom than becomes a theorem proven from the axioms, but one chooses the simplest form for axioms.
Imo the term "collapse" is unfortunate, turning the wavefunction into a balloon. It just means that an instance has been sampled from a probability distribution. The term "collapse" tries to describe, is an unfortunate shorthand, for the fact that after a measurement, the boundary conditions of the quantum mechanical problem have changed,  a new wavefunction will describe the system, not the old one. If a particle decays, the decay products are described by a new wavefunction. If an excited atom emits a photon, ditto. To "observe" at a particle level, means to "interact" and that introduces new boundary conditions.
So imo the parenthesis should not have been there, as it carries too many misconceptions.
