I am not familiar with quantum channels but I know a little about this. You are definitely missing a bit more, namely, that you should also explicitly include the measurement apparatus, separately from the environment. Due to the fact that there is no such thing as a purely classical object, in other words, both the measurement apparatus and the environment are all quantum too, we have to consider that
$$ \mathcal H_o \, \otimes \, \mathcal H_m $$
is an open quantum system, where $\mathcal H_o$ is the Hilbert space of the object, and $\mathcal H_m$ is the Hilbert space of the measurement apparatus. By the purification of quantum states, we know that all such open quantum systems can be seen as a part of a bigger closed quantum system, the missing part we can safely assume to be the environment. i.e.
$$ \mathcal H = \mathcal H_o \, \otimes \, \mathcal H_m \, \otimes \, \mathcal H_e $$
is the relevant complete Hilbert space we are taking as the stage. Here is a good time to discuss a small detour: We usually think of the measurement apparatus as a macroscopic device, in which case we would need to separate the ``pointer states" as macrostates. That would be quite annoying to discuss. Instead, we might follow the original pioneers and consider microscopic systems as the measurement apparatus for simplicity.
Once we have the complete Hilbert space $\mathcal H$, basically everybody can accept that the correct next step is to unitarily evolve the entire object+measurement+environment in time. This causes all 3 things to be entangled with each other.
From here on, there are plenty of interpretation choices. The most natural is to take the decoherence choice (which is common to Many Worlds, Pilot Wave, and quite many others), that the unitary evolution led us to simple object+measurement states (each) entangled with complicated environment states. Because the environment quickly devolves into much more entangled states, this means that undoing the entangling happening in the environment is incredibly unlikely to happen, so that then each object+measurement state essentially will no longer interfere with other such object+measurement states. This has the attractive properties that
- the process of measurement is not mystical
- the process of measurement (including the ``collapse") occurs over finite time
- it has been simulated and results agree with experiments
- the theory is internally consistent
- it explains why modern experiments with tighter and tighter noise controls can see bigger and bigger stuff being put in quantum superpositions, as opposed to Anderson's interesting example in his ``More is Different" paper that after a certain size, the quantum superpositions get washed out.
The mathematically standard thing to do here, is to form the density operator of such a triply entangled pure state, and then do a partial trace over the environment. That then gives a impure state, a mixed density operator over object+measurement. Each pure object+measurement state composing the impure state would evolve independently.
It is when you want to extract the independent pure object+measurement state do you actually impose the projection. Of course, once you apply projection, it will be compatible with all accepted interpretations, since they all agree with the Copenhagen prescription.