The wave function is still symmetric, but it takes on a bimodal distribution, with very little overlap. For any given event, it will be affected only by the half of the distribution that it's in. The other half has basically zero effect. The further time evolves, that effect becomes even smaller -- as in, the odds of an experiment demonstrating it quickly go towards 1 in 10^googol^googol.
You can round that down to exactly zero and call it "collapse". Or you can keep thinking about the entirety of the wave function, and call it a "multiverse". That rounding is technically invalid, but it simplifies the conceptualization (and the math) to a massive, massive degree without affecting the outcome in any pragmatically measurable way.
(One more caveat: "symmetry" implies we're talking about a wave function with a 50-50 superposition. That's not a requirement, but it simplifies an already complex explanation.)
Does this not imply that there is an asymmetry, one half of the state gets imprinted, the other half neglected? This however also raises the question about the basis, what is a superposition and what is not depends on the choice of basis. Is there a special basis just as pointer states are somehow special?
Indeed, as you say, Decoherence explains why certain bases are special: when a system is in a pointer basis state, it does not continue entangling with environment (or, at least, does so minimally). When a spinning particle enters a Stern-Gerlach apparatus oriented in z-direction, spin-z is the pointer basis of the system during its time in the apparatus. A spin-up or spin-down particle does not entangle with the environment, but spin +x state would quickly entangle with environment, placing environment in a superposition and "branching" the total state vector of all the stuff in the universe.
Quantum Darwinism is just a refinement of this picture in which the "environment" interacting with the system is itself modeled a series of fragments (i.e. all the different photons that bounce off object). It turns out that the information about which pointer basis state the system is in (spin up or spin down) is redundantly encoded in each of these fragments. Hence, intercepting one photon that interacted with system and reveals "spin-up" (because the particle is in upper path) agrees with other photons that also bounce off object.
BUT, of course, due to linearity of unitary time evolution, there is another "branch" in which spin-down was the outcome of the measurement and everyone agrees on spin-down. This is exactly the Everett picture.