So let’s say you start a project, get 70% there, it’s version roughly 0.7. The project blows up, people get interested, lots of attention is paid to the periphery. Everyone is now using your project but it’s still missing 30% of what you’d consider necessary to be feature complete. What do you do?

That's the big misunderstanding here, isn't it? That software must be 'done', 'finished', 'feature complete', etc, in order to have major version number greater than 0.

I would (I hope), release version 1.0 soon after I notice people actually using the software, document the existing API, keep releasing new features as minor version updates (1.x), and occasionaly do a major version bump if I see that I must break the existing APIs. As semver prescribes.