It certainly does not seem natural to apply the counting rule and say that there is a 50% chance that you are now Mitchell-1.
Let's suppose I am to be disintegrated as you suggest, to be replaced with these atomic duplicates, Mitchell-1 and Mitchell-2. But we add that Mitchell-1 will be gunned down, and Mitchell-2 will not be.
Now suppose you ask me, before this disintegration, "what are the odds that a randomly selected future duplicate of yours will be one that is gunned down?" The answer is 50%.
But this is analogous to the situation in Many Worlds, before a branching. I, here, now, will cease to be; in my place will be a variety of distinct successors. Why is it illegitimate to reason about them in exactly the same way?
Now suppose you ask me, before this disintegration, "what are the odds that a randomly selected future duplicate of yours will be one that is gunned down?" The answer is 50%.
This is true, but irrelevant. There is no random selection going on. By the same token I could say that if you asked "What are the odds that a future self selected according to the Born rule will observe spin-up in this experiment?" you'd recover quantum statistics. But then you'd rightly challenge me by asking why this particular question should matter. Well, I ...
The subject has already been raised in this thread, but in a clumsy fashion. So here is a fresh new thread, where we can discuss, calmly and objectively, the pros and cons of the "Oxford" version of the Many Worlds interpretation of quantum mechanics.
This version of MWI is distinguished by two propositions. First, there is no definite number of "worlds" or "branches". They have a fuzzy, vague, approximate, definition-dependent existence. Second, the probability law of quantum mechanics (the Born rule) is to be obtained, not by counting the frequencies of events in the multiverse, but by an analysis of rational behavior in the multiverse. Normally, a prescription for rational behavior is obtained by maximizing expected utility, a quantity which is calculated by averaging "probability x utility" for each possible outcome of an action. In the Oxford school's "decision-theoretic" derivation of the Born rule, we somehow start with a ranking of actions that is deemed rational, then we "divide out" by the utilities, and obtain probabilities that were implicit in the original ranking.
I reject the two propositions. "Worlds" or "branches" can't be vague if they are to correspond to observed reality, because vagueness results from an object being dependent on observer definition, and the local portion of reality does not owe its existence to how we define anything; and the upside-down decision-theoretic derivation, if it ever works, must implicitly smuggle in the premises of probability theory in order to obtain its original rationality ranking.
Some references:
"Decoherence and Ontology: or, How I Learned to Stop Worrying and Love FAPP" by David Wallace. In this paper, Wallace says, for example, that the question "how many branches are there?" "does not... make sense", that the question "how many branches are there in which it is sunny?" is "a question which has no answer", "it is a non-question to ask how many [worlds]", etc.
"Quantum Probability from Decision Theory?" by Barnum et al. This is a rebuttal of the original argument (due to David Deutsch) that the Born rule can be justified by an analysis of multiverse rationality.