I think part of why they believe that the Ring is a Middle-earth issue and that those in the West would not receive it is simply because the Valar seem to have made it clear that that's how they operate. In the Silmarillion, the Valar start out very active in Middle-earth, and then they withdraw to Valinor before eventually inviting the Elves to live with them. Eventually, they seem to realize that this wasn't the best idea, and they don't do the same thing when Men arrive. However, rather than get more involved in Middle-earth, they get less (at least, directly and obviously). After Numenor's fall, when the world is reshaped, do the Valar directly interact again? The whole thing seems to lead up to the Valar deciding that it's best for the peoples of Middle-earth to work things out, with the Valar operating behind the scenes rather than in the spotlight. The Wise likely realize that, while the Ring is a big deal, it's not yet the sort of thing that would require direct, divine intervention; and therefore the Valar would not receive it.
It just occurred to me, but in a way, this parallels the discussion we were having last night about Sauron and Morgoth. Sauron is travelling down the same path as Morgoth; and as time goes on, he's slipping into much the same style. Could the Valar be doing something similar, but with Eru as their model? We know that Eru is not completely absent from Middle-earth, however he tends to work subtly and through the actions of others rather than getting directly involved (with a couple major exceptions). Perhaps the Valar are, over time, realizing that they need not be absent from Middle-earth, but that their role should be less direct than they sometimes made it in the old days.