We aren't really given the details of the council held by the Valar. The Maiar's associations with a Valar could mean (and suggested to denote that) they are just a part of that Valar's people.
It also seems that it was sort of on a volunteer-basis. When Manwe asked who would go, only Saruman and Alatar stepped forward.
It's quite possible that Ulmo, looking on, was content with Manwë's selections. He might've known of Olórin's character, or perhaps he just trusted Manwë's judgement enough to think better of imposing another Maia to be sent (as Yavanna did - and we saw how that turned out, didn't we?).
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...take counsel with thyself, and remember who and what thou art.
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