Earlier quoted context omitted.
The point of whiteboarding isn't to "build" correct/functional code - it's to get an idea of your thought process and how you go about solving a problem. It's not perfect (and there are both advantages and disadvantages to using a computer instead of a whiteboard) but when done correctly it gives a lot more useful insight than sitting there asking questions about their past the whole time.
Why is thought process relevant? Different people have different thought processes. I would estimate that recruiters would bias to hire candidates that have similar thought processes to themselves, even if that doesn't correlate well with actual work performance.
Nobody is saying otherwise. The point isn't to make sure your thought process is identical to the interviewers (good interviewers will be more impressed if it isn't), it's to make sure you have one.
That may sound silly but read Jeff Atwood's thoughts on FizzBuzz[1] to realize how many "programmers can't program." Coding exercises are used to make sure you know how to think through a problem and solve it.
> I would estimate that recruiters would bias to hire candidates...
I'm not sure if you used the wrong word or not, but technical interviews are usually done by engineers or engineering managers, not recruiters. But yes, if I were interviewing somewhere and I sufficiently solved a problem but they didn't like it just because my thought process was different I would consider that a red flag.
[1] http://blog.codinghorror.com/why-cant-programmers-program/