And bias can certainly be of different magnitude. A weighted dice can have a bias that shows up after only 10 rolls and another one can only show it's bias after 1000.
A random sample is not a product of a biased sampling method, but it can be (and almost always is) biased, it's just if you take enough such samples, the biases will tend to offset, which is why we say the sampling method isn't biased.
I think that this hypothetical example would be evidence of bias. "woman:doctor :: man:nurse" as an analogy only works because of gendered implications of 'doctor' versus 'nurse', so if that were a robust output of the model it would imply that the word2vec found that gender was of great significance in the manifold near 'doctor'. (The analogy would just be applying a negative weight to the gender basis, but that's mathematically fine even if not customary.)
I think it is also important to note that this is an argument over which of two methodologies is best. The answer likely is that it depends on the use case. Researchers are not being accused of using underhanded methods, manipulating data or cheating.