No sign of a Heisenberg cut has been observed so far, even as experiments involving entanglement of larger and larger molecules are performed, which makes objective-collapse theories hard to consider seriously.
Bohmian theories are nice, but require awkward adjustments to reconcile them with relativity. But more importantly, they are philosophically uneconomical, requiring many unobservable — even theoretically — entities [0].
That leaves either many-worlds or a quantum logic/quantum Bayesian interpretations as serious contenders [1]. These interpretations aren't crank fringe nonsense. They are almost inevitable outcomes of seriously considering the implications of the theory.
I will say that personally, I find many-worlds to focus excessively on the Schrödinger-picture pure state formulation of quantum mechanics. (At least to the level that I understood it — I expect there is literature on the connection with algebraic formulations, but I haven't taken the time to understand it.) So I would lean towards quantum logic–type interpretations myself.
The point of this comment was to say that many-worlds (or "multiverses", though I dislike the term) isn't nonsense. But it also isn't exactly the kind of sci-fi thing non-physicists might picture. Given how easy it is to misinterpret the term, however, I must agree with you that a self-aware science communicator would think twice about whether the term should be included, and that there may be not-so-scrupulous intentions at play here.
Quick edit: I realise the comment I've written is very technical. I'm happy to try to answer any questions. I should preface it by stating that I'm not a professional in the field, but I studied quantum information theory at a Masters level, and always found the philosophical questions of interest.
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[0] Many people seem to believe that many-worlds also postulates the existence of unobservable parallel universes, but this isn't true. We observe the interaction of these universe's every time we observe quantum interference.
While we're here, we can clear up the misconception about "branching" — there is no branching in many-worlds, just the coherent evolution of the universal wave function. The many worlds are projections out of that wave function. They don't discretely separate from one another, either — it depends on your choice of basis. That choice is where decoherence comes in.
[1] And of course, there is the Copenhagen "interpretation" — preferred among physicists who would rather not think about philosophy. (A respectable choice.)
