This reminds me of religious arguments about how evolution can't account for complex organs like eyes. I think in general humans just have a very difficult time conceptualizing massive numbers and systems, and as such the emergent phenomena that come out of it seem unreal.
Whenever I look into neuroscience, it looks like researchers are getting a clearer and clearer idea of how things work (as you noted, we even understand how to manipulate things at a basic level), but the system is so massive that it's slow going. As such, there's a certain "god of the gaps" argument that people turn to, where anything that researchers haven't yet fully understood gets treated as evidence that there's "something more." You see the same sort of arguments for religious explanations for other natural processes that are still being understood.
When I listen to people who actually study this (and make progress on it), they don't see to hit any limits using a physical approach. The "hard problem" seems to mostly exist in the minds of people outside the field who are unsettled by the thought that we might not be special (a pretty common reaction whenever science makes progress on subjects).