The stable proportion is seen around the world, including in places like Germany where they are doing tons of testing and certainly far more than just those who turn up to hospital.
You're right that it's not a uniform sample, but the consistency across the world is very suggestive. Not everywhere has test shortages.
We're not testing the general population. Its testing of folks coming to a hospital? Folks who aren't having hospitalizing symptoms are not tested at all, and sent home to quarantine.
So if tests doubled that means that folks are feeling bad at an increased rate. Showing up at a hospital with alarming symptoms.
You can finagle the statistics both ways - by ignoring what the 'test sample' is and assuming its uniform for instance.