Yes, if your variables are perfectly linearly dependent they get dropped. Did anyone say otherwise? I did not think about this case because most correlated measures causing multicollinearity problems aren't perfectly 'linearly dependent'. Linearly dependency usually only comes up practically if you miscoded some of your independent dummy variables (e.g. adding both 'male[0,1]' and 'not male[0,1]' as two categorical predictors). So I am not really sure of your point.
As to your second point, it might be unbiased but the statistical inference (i.e. p-value) would be incorrect with multi-collinearity..thus again, I am not sure of your point when you are only repeating what I said.
Moreover, it may not be particularly meaningful to the researcher even if the parameter estimate is unbiased. One frequently finds with multicollinearity that the signs of effects will switch (- to +, or + to -) as you add highly correlated predictors into a model, in oft-theoretically questionable ways, but does serve to remind one that the parameter estimates are only meaningful in the context of the other predictors in the model.