While I lean the same way, the measurement problem is definitely harder than you say. If it's just entanglement with an external system, you still have to answer how that superposition ends up with you personally observing only one result.
You, however, being a large, warm thing decohere rather rapidly, even if the formal measurement apparatus can be made not to. Your consciousness is a classical phenomenon, so you can only perceive one possible classical outcome. If you believe in Many Worlds (which is the most natural interpretation to pair with this resolution of the measurement problem, but is not absolutely forced), then there are other versions of you out there in different parts of state-space which perceive different outcomes, and may even be observable as elements of a superposed-you in principle if not ever in practice, but there is no means of transferring information between any of these different versions of you.
EDIT: To whomever downvoted the parent... why? It is, after all, a perfectly reasonable question, especially (although I do not presume that knzhou personally holds this view) in light of the reasonably common assumption that consciousness is somehow quantum in nature itself.