I spent a good bit of time trying to explain this to laycreatures
Sounds like the blind leading the blind.
You can't naively apply Popper in this case (who in any case is by no means the last word on philosophy of science), because the situation is quite complex: the article describes a possible consequence of existing established theories, including quantum field theory, general relativity, and Big Bang cosmological models. As such, Popper's rules don't say anything about those theories not being science, or whatever.
While it's true that "the math does not lead only and exclusively to that conclusion", it's a valid possible conclusion. As such, given the status of the theories that it's based on, we can't avoid taking it seriously as a possible description of reality. The task then becomes to discover if there's any way to improve our certainty about its correctness or lack thereof, and that's why people like Linde write papers about this stuff. Rejecting this as "not science" or whatever based on one particular view of what science is, is terribly short-sighted, and it's lucky that actual scientists don't pay attention to such nonsense.
One of the interesting consequences of eternal inflation style theories is that in principle, it addresses questions of fine-tuning. One can take the "evidence of fine tuning" as an argument in favor of multiverses in some form. From that perspective, the idea that our observable universe that started with the Big Bang is the only universe is actually the more difficult theory to defend, since we don't know how some of the parameters managed to come out on the knife-edge of allowing the universe to expand to a useful size and have useful properties like the ability for matter to form.
Re Popper, you should look into Imre Lakatos, who pointed out various flaws with basing all of science on falsificationism. See e.g. The Methodology of Scientific Research Programmes.