As far as I can see, they are simply arguing this: if one component of the wave function, namely the one that is actually measured at first, is in this world, then it must be that all other components are also. No doubt an MWI fan could argue against this, because the reality of this world is just as debatable as the reality of the many. This is philosophy, not physics.
There are alternative views, which disagree with most MWI and anti-MWI arguments. Many physicists prefer an anti-realist view of QM, but it is difficult to find philosophers who don't wince at a physicist's idea of what anti-realism is about. Most physicists are simply thinking that measurement is paramount, and our classical picture of an a priori world disagrees with experiment. But for true anti-realism one requires more: that the observer's world is itself defined by the observer's choices and experience.
So there are three basic options here: 1. many concrete worlds (MWI) 2. one concrete world (eg. Bohm) 3. no concrete worlds (anti-realism). In the end, since everyone is talking about the same mathematical QM there is no way to resolve the debate physically, without constructing a new theory that relies on one and only one of the above options.