I think that the idea that our personal experience "branches" at every moment to produce multiple universes in which alternate selves exist is perhaps a stretch, i.e. beyond the realms of testable physics, firstly.
That isn't what the Everett interpretation says. It says that when a quantum system in a superposition of mutually-orthogonal states interacts with (is observed by) a quantum observer, the observer enters a superposition of mutually-orthogonal states correlated with the original system. Because they are mutually orthogonal, they do not interact, and cannot see one another. The observer is a superposition of observer states, each seeing exactly one outcome, each one apparently randomly selected. To the observer, it is as if each possible outcome of the quantum experiment happened in a separate world. But no universes are actually created, or 'split'. There is only one universe. No additional universe-splitting mechanisms or 'new physics' are introduced. It is simply the standard unitary evolution of wavefunctions from standard quantum physics, that predicts observers in a quantum universe will see exactly what we do see. There is no need for wavefunctions to collapse to explain why we see only one outcome. Mutual invisibility of orthogonal sub-states already explains it.
Proponents like it because of its elegance and simplicity, because it has much nicer properties like being linear, deterministic, local, causal, realist, etc., and because it constitutes a more detailed and comprehensive explanation of things. But because it makes exactly the same predictions about what is observed in Copenhagen collapse, the difference is not testable. The differences all occur in the part of the universe that Everett says we can never see; that Copenhagen says vanishes.
So, if many worlds is not falsifiable in the original context, could it at least approach falsifiability in the context of the insanely huge number of conscious entities that are "said" to exist in this reality?
Not for this reason. 'Insanity' implies conclusions drawn from invalid or disordered reasoning - and the number of conscious entities is a straightforward logical consequence/implication of the axioms of quantum physics excluding those describing wavefunction collapse. What's insane is rejecting the consequences of valid reasoning purely because the result is unfamiliar, unintuitive, or has emotionally undesired implications. It's no more 'insane', merely for being a very big number than the number of atoms in everyday objects, or the number of possible states of all the air molecules in a room.
Since there can never be any experimental evidence distinguishing them, you are free to accept or reject either on aesthetic or utilitarian grounds. It's far easier to do calculations assuming collapse. The picture of the state of the universe at any given moment it evokes is far easier to imagine, and far closer to our classical physics view. But collapse propagates faster than light, backwards-in-time, by an undefined, unmeasurable, unobservable, and really hard-to-imagine or understand mechanism. Neither interpretation is at all comfortable to our classically-trained intuitions.
However, both Copenhagen and Everett are still falsifiable in that they both make the same predictions about observations, that could yet turn out to be untrue. They would both be falsified simultaneously by such evidence that our current understanding of quantum physics was wrong. And given the well-known problems integrating quantum physics with general relativity, it wouldn't even be very surprising.
