There's no clause in there about your consciousness getting magically teleported to worlds in which you survive.
You are absolutely correct - there is nothing about the MWI that implies such a thing. Such a thing is implied by consciousness itself.
Here's a thought experiment. You implement a conscious system on a computer. After letting it run for a year, you interrupt the software but save all of its state. After waiting 20 years, you restart the software from the same point on a completely different computer running an emulator for the old system.
The consciousness, whose reference of time is entirely independent of the real world, does not "feel" the 20 year pause. It "magically teleports" to the new computer.
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You are absolutely correct - there is nothing about the MWI that implies such a thing. Such a thing is implied by consciousness itself.
Here's a thought experiment. You implement a conscious system on a computer. After letting it run for a year, you interrupt the software but save all of its state. After waiting 20 years, you restart the software from the same point on a completely different computer running an emulator for the old system.
The consciousness, whose reference of time is entirely independent of the real world, does not "feel" the 20 year pause. It "magically teleports" to the new computer.