While true that the Valar did care about what took place in Middle-earth, they too seemed disclined to drastic intervention. The Istari, while sent to aid the fight against Sauron, were forbidden to use their full power to defeat him independently, their role being to encourage the Free Peoples to fight him themselves. And the Valar only arrived to fight Morgoth after Earendil's pleas. So while they may be troubled by Sauron's theoretical domination, they don't seem likely to send a force to 'clean house,' as it were.

Maybe they would retry sending emissaries like the Istari (though that plan initially had a 4/5 failure rate), but not directly coming to battle with Sauron.
As to Eru plans ultimately working in the end, I'm sure that they would have led to Sauron's eventual defeat, but only after considerable destruction and heartache. How it would have happened, I can't say.
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Of course the argument can be put forth that Eru Ilúvatar left the free peoples of Middle Earth to govern their own affairs since many of the Elves had originally refused the summons of the Valar and chosen to dwell in the Mortal Lands. Yet this does not apply to races such as Hobbits whom were presumably created as the Children of Ilúvatar and are vulnerable. Subsequently they are never mentioned to have inhabited Valinor or had any ties with it (open or secret). They never received such an offer from the Valar.
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The ability to govern their own affairs isn't so much related to the Elves refusing the Valar's summons as to an intrinsic gift from Eru. In fact, I think that the more vulnerable races were actually the ones given more freedom of choice. Being at work, I don't have my books with me right now, but I think there's a passage about Men (and Hobbits, by extension) being the only Children not bound by the Fate set by the Music at the beginning. Actually, my personal theory would be that the more choice available, the less willing the Valar might be to intervene, since they would be less sure of Eru's plans, though I'm not sure I can back this up textually.