In short, I wouldn't consider "your whiteboard code is missing a semicolon" to indicate a lack of productivity overall, only a lack of practice coding on a whiteboard. (It surprised me how much better my whiteboard code got over the course of a quarter of teaching.)
For more complicated tasks I guess you can say it's debatable when whiteboard is OK and when you should give the candidate access to a computer.
And that's under ideal conditions! Writing code on a whiteboard is so far from ideal for most people that keeping your variable names consistent is a challenge — balancing parentheses and brackets and keeping track of little dots is outright grueling, and doing all this during a job interview is enough to make even very competent programmers choke.
Give me a laptop with just a text editor and watch over my shoulder.