What would have he gained if he did so?

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"Even darkness must pass. A new day will come. And when the sun shines it will shine out the clearer."

I believe the elves, or at least the lords of the Noldor, knew what the Wizards were because Cirdan met them when they sailed East. The elves are also the only ones that probably remember much about the First Age and the interacts with the Valar. Going into Bree or the Shire and saying "hey guys, I'm a messenger of the Valar come to help you battle evil" probably would have not done much for his cause since no one would understand what he was on about.

Also, I got the feeling from the Fall of Numenor that the Valar were basically done with Middle Earth and, win or lose, the Wizards were the last token of assistance the world was getting from them. It wouldn't make much sense for Gandalf to talk up his connection with the Valar at all because he'd eventually have to explain that there was no more help coming and the "gods" had basically abandoned everyone to Sauron. Which, in turn, would not do much to inspire people to resist Sauron.

In sum, I think Gandalf didn't reveal he was a Maiar because it wouldn't have help him accomplish his mission.

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"Draw your sword with a heavy heart, but swing it with a heavy hand"