Branching universes need not be falsifiable to be taken seriously, because that's not the theory. Rather it's a consequence of a theory (evolution of Schrödinger's equation sans collapse) and that theory is very falsifiable. In order for a theory to be falsifiable, you don't need all aspects of the theory to be falsifiable, just one. And branching universes is a prediction of a theory that has experimental support.
Let me take a non-quantum example to make the argument in a more intuitive setting. If we send out a spaceship at a high speed away from Earth, due to the ongoing expansion of the universe eventually the spaceship will be far enough away that the space between Earth and the spaceship will be expanding faster than the the speed of light, and hence the future light cones of the Earth and the spaceship will not intersect. There's no experiment we can do on Earth that could give us information about the spaceship once the spaceship hits this point of no return, but we still take seriously the proposition that the spaceship is still "real" and doing well because we have a broader theory (i.e. all of physics) that predicts it, even if we can't test this specific prediction.
(I view these two scenarios as being superficially dissimilar, but under the surface fairly similar. In each there is some element of "realness" that is predicted by theory but inaccessible to measurement. Only one is intuitive and the other is not.)
Putting aside a recent claim that the universe's expansion isn't really accelerating - and instead someone forgot to plug spacetime dilation into the cosmological equations - we're still left with the issue that once the space probe leaves our light-cone we can't know that it wasn't hit by an asteroid.
That's of course not the point of the thought experiment. Just the mass-energy and "realness" of the spaceship after it leaves causal contact with Earth is enough.
Consider the statement:
"Once any object gets far enough away from me that our future light cones do not intersect, it ceases to exist."
And compare with the statement:
"Once any configuration gets so far away from my configuration in configuration space that I can't experimentally detect it, it ceases to exist via waveform collapse."
The Copenhagen Interpretation is claiming something akin to the second statement, but that seems to me to be logically isomorphic to the first statement, of which our intuitions are much better at grasping the problems.
Let me take a non-quantum example to make the argument in a more intuitive setting. If we send out a spaceship at a high speed away from Earth, due to the ongoing expansion of the universe eventually the spaceship will be far enough away that the space between Earth and the spaceship will be expanding faster than the the speed of light, and hence the future light cones of the Earth and the spaceship will not intersect. There's no experiment we can do on Earth that could give us information about the spaceship once the spaceship hits this point of no return, but we still take seriously the proposition that the spaceship is still "real" and doing well because we have a broader theory (i.e. all of physics) that predicts it, even if we can't test this specific prediction.
(I view these two scenarios as being superficially dissimilar, but under the surface fairly similar. In each there is some element of "realness" that is predicted by theory but inaccessible to measurement. Only one is intuitive and the other is not.)