Impossibility of deriving the Born Rule?

Question

Background

Let's say I have a physical system. When a measurement happens we apply the Born rule to say which outcome is likely.

1. Now, while a lot of physicists are of the opinion the measurement is a non-unitary transformation.
2. They are also of the opinion that the whole system undergoes unitary transformation (while a sub-system does not).

This seems to mean to me that the real problem is getting the apparatus to be of the same initial state in both experiments. If one does so then one can evade the Born rule.

Heuristically:

If we consider the physical system alone:

$$ |\text{system in A} \rangle + |\text{system in B}\rangle \to |\text{system in A}\rangle$$

However, when we consider the measuring apparatus as well:

$$ (|\text{system in A} \rangle + |\text{system in B}\rangle) \otimes |\text{apparatus in C} \rangle \to |\text{system in A} \rangle \otimes |\text{apparatus in D} \rangle $$

Note in the first equation the we assume we say the measurement is non-unitary transformation in the second we say the complete system obeys unitarity.

Question

If the above notion is correct the Born rule seems to be an artefact of some notion of randomness of when (the time) the measurement is being done. Is there any physical system within which we time the measurement such that this possibility can be ruled out?