> Do thousands of incompetents also machine-gun apply

Enough that it's HR or some automated software that first screens your application.

> Something isn't adding up.

Yes. The pipeline "posting > application > screening" is now completely broken. In your case it's quite possible it's HR and screening software that your resume is not convincing. I have been hoping for anecdotes and studies in this direction (from people who have access to the HR and/or software screening and are inclined to report) - but it's at least not common. What we do hear from, is tons of people who can program and who are not getting even first interviews.