We don't need anthropic reasoning under MWI in order to be surprised when finding ourselves in worlds in which unlikely things happen so much as we need anthropic reasoning to conclude that an unlikely thing has happened. And our ability to conclude that an unlikely thing has happened is needed to accept quantum mechanics as a successful scientific theory.

"We" is the set of observers in the worlds where events, declared to be unlikely by quantum mechanics actually happen. An observer is any physical system with a particular kind of causal relation to quantum states such that the physical system can record information about quantum states and use the information to come up with methods of predicting the probability of previously unobserved quantum processes (or something, but if we can't come up with a definition of observer then we shouldn't be talking about anthropic reasoning anyway).

1. According to MWI, the (quantum) probability of a quantum state is defined as the fraction of worlds in which that state occurs.
2. The only way an observer somewhere in the multi-verse can trust the observations used that confirm quantum mechanics probabilistic interpretations is if they reason as if they were a random sample from the set of all observers in the multi-verse (one articulation of anthropic reasoning) because if they can't do that then they have no reason to think their observations aren't wrong in a systematic way.
3. An observer's reason for believing the standard model of QM to be true the first place is that they can predict atomic and subatomic particles behaving according a probabilistic wave-function.
4. Observers lose their reason for trusting QM in the first place if they accept the MWI AND are prohibited reason anthropically.

In other words If MWI is likely, then QM is likely iff AR is acceptable.

I think one could write a different version of this argument by referencing expected surprise at discovering sudden changes in quantum probabilities (which I was conflating with the first argument in my first comment) but the above version is probably easier to follow.