Possibly, if one is pedantic, the phrase "unbiased opinion" can be seen as an oxymoron.
But if one is attempting to communicate with an audience, and to do so succinctly and without excessive verbiage, then I believe that the phrase does its job.
The message being conveyed, for those who would rather be argumentative than share communicative clarity, is that we will state our opinions in a manner that is not biased by external factors. For example, if I write that Canon's latest camera is simply a warmed over version of a model from three years ago, this is an opinion. It is neither objectively right not wrong, it is just an opinion.
Is it biased? I would maintain that since this opinion is not influenced by anything other than my personal evaluation of the marketplace, it is unbiased. No influence other than my own thoughts have lead me to this conclusion.
The purpose of written communication should be to convey thoughts succinctly and unambiguously. If the phrase "unbiased opinion" does not do that, then fine, let's discuss alternative wording. But otherwise the jibes are simply seen by me as pedantry.