[Inziladun]

I think the example of the Istari you give is enough of an indication that the Valar (and by proxy, Ilúvatar,) "cared".

No, the Valar did not choose to directly confront Sauron, and that was a deliberate effort to avoid past mistakes. They thought the defeat of Sauron important enough, though, that they asked their own servants, the Maiar, to become truly embodied peers of the afflicted Children of Ilúvatar and commit themselves to a long-term exile in Middle-earth, enduring its pains, discomforts, and dangers.
In order to do this, the Valar had to ask, and did receive permission from the One. It could be argued that Ilúvatar himself put the idea into their heads, or at least knew that it would occur to them.

The "caring" of the Valar was directly connected to the "caring" of the One, for the Valar were merely the Governors of Arda, not themselves rulers. They were charged with overseeing the world and accomplishing Ilúvatar's will, making them the "hands" to his "brain".

The larger theological issue is a question of why the One allowed evil to exist, and what was the connection to his Music. Good luck with that one. If one accepts Eru as the Creator of Arda, one must also accept his carte blanche to order his creation as he willed, though I think that his ultimate benevolence is in evidence in the books.