You've said it more eloquently than I can, but I want to bring up that:
"results in many false negatives..." is not some universal truth.
Is there evidence that shows current software interviews turn down way too many great candidates?
There should exist a large enough group of developers who have been rejected a number of times[0], who have gone on to be very successful[1] in their careers. Does this data exist?
I agree that the current process isn't perfect, and it does produce some false negatives and some false positives. The amount which, has yet to deter me from beliving that the system works, as intended. Certainly there are some fringe candidates, who are turned away, but I don't think it's as onerous as people are making it out to be.
Also, most companies didn't start out with unlimited piles of cash, nor could they have their pick of employees. They had to work at it, and I don't believe the culture of their hiring has changed significantly.
For the record, I struggle with traditional coding puzzle interviews. But I also know coding on the whiteboard is a skill, which can be practiced, and with time, improved upon, such that if you are a really good developer, but struggle with passing interviews, you can deal with that problem by becoming better at interviewing.
Or you know, create something awesome, and never have to worry about interviewing again.
[0] Failing one or two interviews isn't really indicative of anything. So this group would require consistently failing multiple standard coding interviews.
[1] How to measure success? I say gone to be a major contributor to the success of a company, or built their own company that is considered "successful."