this process depends on me subjectively experiencing the result of the experiment.
All results of the experiment will be experienced by a future version of you.
MWI has to add a postulate that some mechanism placed me in a certain branch of the possible outcomes.
No, this is definitely the Vertiginous Question. The "mechanism" that puts you in a certain branch is the same one puts you in a certain body. Are you also going to demand that neuroscientists answer the Vertiginous question before they can say that other people exist?
you need to add that postulate.
That "postulate" already exists if you believe in consciousness in the first place.
“you actually observed all outcomes” is just factually wrong from an empirical viewpoint.
Literally the opposite: empiricism requires objectivey, trying to insist that only the things that you personally subjectively experience counts is as far from that as you can get.