The proof is my own personal experience. If programmers were truly scarce, then every time a programmer passes a programming puzzle interview, then they'd be offered a job. I can't tell you how many interviews I've aced over the years only to be ghosted afterwards. If ghosting existed (which is very much does, and is quite common), then that means the market is oversaturated.
The experience for the candidate is that they make it through all the gates and then get ghosted while the hiring manager is stalled by their bosses to approve the hire. That has happened to me often enough in the past three years that I now just think of interviews as an opportunity to meet other nerds.
That doesn't mean the job market is not getting more competitive - it might well be, but that does not mean it's oversaturated yet, not even saturated.
I really don't think being able to pass these "programming puzzle" type interviews is a good measure of a good engineer. Just finishing them is often not what a company is looking for.
Even as a quality candidate there might be a lot of way that you are going wrong. I would advise you try and get your CV looked at. This can make a big difference. I'm sure there is lots of advice out there. What market are you looking in?