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?