I think your argument, on the other hand, relies on the assumption that the reviewers are objective and rational. In an ideal world, they would be. In practice, however, just seeing a conclusion they don't agree with may be enough that they won't evaluate the argument wholely on its merits.
As a somewhat over-the-top example: suppose you were a reviewer in a popular biology journal which, due to its popularity, also got lots of low-quality submissions. Now you are told to review an article whose author you don't know, but which argues that our common conception of how evolution works is actually really flawed. You know that lots of deluded creationists bombard the journal with submissions arguing points like these, and you have had the misfortune of personally reviewing several before this. (They were all completely worthless.)
Then an alternate universe where the situation is exactly the same, but the journal hasn't implemented a policy for double-blind reviews. You are told the name of the author, which is Richard Dawkins.
In which scenario do you think you'll give the submission a more fair hearing?
Previously, Robin Hanson pointed out that even if implementing anonymous peer review has an effect on the acceptance rate of different papers, this doesn't necessarily tell us the previous practice was biased. Yesterday, I ran across an interesting passage suggesting one way that anonymous review might actually be harmful:
(From Richard A. Posner, "Aging and Old Age")
If this hypothesis holds (and Posner admits it hasn't been tested, at least at the time of writing), then blind review may actually slow down the acceptance of theories which are radical but true. Looking up the Peter Messeri reference gave me the article "Age Differences in the Reception of New Scientific Theories: The Case of Plate Tectonics Theory". It notes:
In this light, an older scientist's acceptance of a new, radical hypothesis should tell us to give the new hypothesis extra weight. It might even be appropriate to apply a heavier "reputation weighting" on controversial theories than established ones - we'd expect the established scientist to write papers supporting the established theories, but not controversial ones. However, blind review makes this impossible. The reviewers may even mistake the established scientist as an undiscriminating free thinker, who endorses any controversial theories simply because they're unpopular. This will slow down the acceptance of hypotheses which are, in fact, correct.
A possible objection to this could be that old scientists are unlikely to change their minds, and therefore old scientists not getting enough credit for their achievements won't have much of an effect in the spread of new hypotheses. (After all, if no old scientist endorses controversial hypotheses, then it doesn't matter if those endorsements aren't properly weighted in review.) Not so. Messeri finds that science does actually progress a lot faster than "one funeral at a time" (as Max Planck put it), and old scientists are ready to adopt new theories given sufficient evidence. While the last holdouts for outdated theories do tend to be the old, their increased security also makes them the first who are willing to publicly support new theories: