I believe the “advanced civilization VS closed-doors science” problem you propose is inherently faulty. The final step of the scientific method is “share your results”, and they did that. It cannot be accessed anymore, but they attempted to share it and it benefited their people at the time. However, closed-door science benefits no one.
Take, for instance, the Greco-Roman mix for concrete. They produced that scientifically, repeating experiments with their volcanic ash-filled cements to make the best they could, and shared those results across their territory. While that knowledge was lost for centuries, it was still science.
However, a cement company producing a better product and not sharing the production is not science. It’s an economic pursuit. Its goal is not to help everyone to produce a better product, but to produce a better product and hide the results. Even if it uses most of the scientific method, if it doesn’t share its result its not science.
And no, monetary stake isn’t the only reason someone may not be doing science properly. A major one we see time and time again is the desire for science-based renown.
For instance, Einstein is renowned for being one of the smartest men alive, who shaped a lot of our modern understanding of the universe. He may ultimately be wrong, but we have yet to prove that and in fact continue to prove he was decades ahead of his time.
Contrary to Einstein though is Edison, while he did do quite a bit of real science, he was less focused on actually advancing our knowledge and more focused on his own image. For instance, he invented the electric chair solely to try and slander Tesla’s AC, and tortured animals in public to try and breed fear.
Additionally, I will throw my hat in the ring and say most of modern academia doesn’t pursue science, as they push for ever-restricting holds on scientific papers in the pursuit of financial gain. While simultaneously depriving the actual researchers any of that gain, money which could surely help raise a lot of institutions into doing more effective research. Instead, they profit off of and restrict the free flow of knowledge to line their pockets, disabling millions of researchers worldwide from accessing it and imposing harsh prices on non-academics who wish to learn.
In the end, when it comes to what is and isn’t science, I will posit a question. If you never heard about it, and can’t reproduce it yourself/watch someone reproduce it, did it really happen? Or has “science” evolved from the pursuit of knowledge to a faith? If we blindly trust the people in power that they are right, and punish those who question their assertions, it can only end in destruction.