This need have nothing to do with conciousness. The measurement problem in quantum mechanics always and only happens when a system we can model with quantum mechanics interacts with a system which we cannot model with quantum mechanics (because it's too complex and/or we cannot know its initial state). The result of this looks like probabilistic collapse of the wave function, but that may be just an artifact of not being able to track the dynamics anymore. One explanation for what happens is to privilege conciousness on the part of the observer, and say that conciousness is responsible for the collapse, but there's no actual need to do so, and in my opinion no good reason to either.
Ok. Your theory says there's never a collapse. Rather, an experimenter and his experimental apparatus become entangled, and the wave function describing both continues to evolve unitarily. When then, according to your theory, does the experimenter become aware of the result of his experiment, and which result does he become aware of, out of the many that are possible? It is undeniable that experimenters do become aware of experimental results. Shouldn't any theory that purports to solve the measurement problem be able to explain how this happens?
I don't mean "how" in any deep philosophical sense. Just that the theory should at least acknowledge that it does happen, and should tell us when. A theory that involves only wave functions evolving says nothing about conscious observers, whose observations it supposedly is trying to explain. How could that possibly work?
Collapse serves two purposes in the current theory. It tells us what wave function to use for predicting future results, namely the collapsed wave function rather than what it was before the collapse. But it also tells us when the experimenter observes this result, namely when the collapse occurs.
You can get around the first point by saying that, to predict a future result, a second experimenter can use an uncollapsed wave function describing both the first experimenter and his apparatus, instead of using a collapsed wave function describing just the apparatus. But how do you explain why the first experimenter observed the result that he did, or any result at all for that matter?
A theory can't explain conscious observation if it refuses even to talk about it.