I wouldn't call Everett 'Anthropic' per se. I consider it an application of the Generalized Anti-Zombie Principle: Here you've got this structure that acts like it's sapient†. Therefore, it is.
As for BM formally specifying how the reality fluid works... need I point out this this is 100% entirely backwards, being made of burdensome details?
Would you say that acknowledging the Born probabilities themselves 'comes out of the blue', since they aren't derived from the Schrödinger equation?
The Schrödinger Equation establishes linearity, thus directly allowing us to split any arbitrary wavefunction however we please. Already we can run many worlds side-by-side. The SE's dynamics lead to decoherence, which makes MWI have branching. It's all just noticing the structure that's already in the system.
Edited to add †: by 'acts like' I mean 'has the causal structure for it to be'
The Schrödinger Equation establishes linearity, thus directly allowing us to split any arbitrary wavefunction however we please.
But many of the more-general lagrangians of particle physics are non-linear, in general there should be higher order, non-linear corrections. So Schrödinger is a single-particle/linearized approximation. What does this do for your view of many worlds? When we try to extend many worlds naively to QFTs we run into all sorts of weird problems (much of the universal wavefunction's amplitude doesn't have well defined particle number,etc). Shouldn't we expect the 'proper' interpretation to generalize nicely to the full QFT framework?
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.