[feanaro]

I think there is some misunderstanding here based on the wording used. OPs point is simply that there is no separate, explicit branching process. Instead, there is only the normal, continuous evolution of the wavefunction.

Our experience is recovered from this by positing that subjective/phenomenological experience is somehow tied to the individual components of the wavefunction. Since the individual components don't interact with each other, it gives the appearance of branching. This is compatible with our observations.

[ithkuil]

Furthermore it firmly places ourselves and our own experiences within the very same fabric of reality it's describing. This is a rather inevitable problem of a fundamental theory of reality, since we ourselves are part of it and our inner working (whatever that is) must be layered on top of that fundamental machinery.

[kgwgk]

So we get from the “the wave function somehow collapses” to “the phenomelogical experience is somehow tied to the individual components wave function” which are somehow connected go the measurements.

[feanaro]

You're right regarding the first part, we don't have a clear explanation of how the wave function might be related to subjective experience, but that's mostly because we don't have a good account of subjective experience at all.

For the second part, there's nothing mysterious going on at all: measurement is simply physical interaction.

[kgwgk]

What are "the individual components of the wave function"? They are a mathematical construction. The wave function can be decomposed in many (and infinity of) ways. Why is one decomposition chosen and not another? Or are all the decompositions equally "(un)real"? The issue is not so simple and it's related to the "mysteriousness" of the first part.