Why is so much emphasis placed on the Ring instead of the Palantiri?

Question

The One Ring provides great power, but its power is primarily to dominate others. Sauron already dominated, it is implied, nearly all of Middle-earth without the Ring. It is implied or stated at various points that with his limitless armies of vassals, he would be able to easily overwhelm the holdouts, again with no need for the Ring. Sauron originally made the Ring to specifically control the leaders of his enemies, but the Ringwraiths had already fallen to him, and the Elves would just remove their rings if he tried, as they had done previously.

By contrast, the Palantiri seem far more powerful. The Palantir that Sauron most likely had, the Ithil stone, could only look in a fixed direction, but it already would have permitted him to see most things along that line, with sufficient concentration. By contrast, had he obtained the Osgiliath stone or the other omnidirectional stone, he would have been able to see anywhere in Middle-earth, at an unlimited distance, eavesdrop on the communications of other Palantirs (in the case of the former), and possibly try to dominate the other users. The applications of such knowledge, especially in conjunction with the mental communication abilities of Sauron as a Maia, would be considerable, and it would be particularly bad for Sauron to have these capabilities if the Ring were not destroyed. Imagine Sauron being able to spy on Elrond's councils, or to send an assassin to kill one of his enemies who would know exactly how to slip past any protection. Indeed, his Eye (probably largely aided by the Palantir in his possession) was a significant threat even without the Osgiliath-stone.

Why, then, did both Sauron focus so much of his efforts on recovering the Ring, and his enemies so much on keeping it out of his hands, relative to the Palantiri, given the power of the latter?

Answer 1

The idea that the Council and Gandalf were distracted by the Ring and had not entirely thought out the true power of the Palantíri is suggested by Tolkien in his essay titled "The Palantíri"

But, although Gandalf had in latter years enlarged his own and the Council’s knowledge of Gondor’s history by study of its documents, his and their chief concern was still with the Ring: the possibilities latent in the Stones were not realised.
_{The Unfinished Tales, Part Four, III: The Palaníri}

Gandalf was concerned with the Ring and had not taken into thought the effects that might occur of Sauron possessing a stone and someone else peering into it. It was not until they'd been shown a demonstration of it on Dol Baran, that the idea became clear. It was with great haste that Gandalf and Peregrin then rode to Minas Tirith, for numerous reasons, undoubtedly hastened by the fear that Denethor had used the Anor-stone.

In Sauron's designs however, he did not regard the stones highly. While he saw value in their use to dominate and bend the will of others, of the two beings he encountered through the stones he was only successful in one case. Furthermore, Sauron did not have the resources to remain under constant watch of the Ithil-stone, and would never allow a subordinate to relieve him.

It must also be considered that the Stones were only a small item in Sauron’s vast designs and operations: a means of dominating and deluding two of his opponents, but he would not (and could not) have the Ithil-stone under perpetual observation.
_ibid.

The stones themselves never suggested to Sauron a greater purpose for his victory. They were not of exceptional interest to him and other than the stone he possessed only two other (lesser) stones remained.

Of the two stones, one remained in the hands of a person who had more right to them by him. Although a Steward and not of a kingly line, in Gondor it was the Stewards who were tasked with caring and using the stones, and after the role became hereditary, their claim to the stones and power through them grew in greatness.

In the case of Denethor, the Steward was strengthened, even against Sauron himself, by the fact that the Stones were far more amenable to legitimate users: most of all to true ‘Heirs of Elendil’ (as Aragorn), but also to one with inherited authority (as Denethor), as compared to Saruman, or Sauron.

Finally, to evaluate what effectiveness Sauron would've had with the stones, while it is true that rather small details could be seen, it required great concentration and will power to enlarge the mostly hidden images, to the extent of spying on a group talking it would be nigh on impossible to concentrate on so many minute images at once.

Great concentration might even enlarge some detail that interested the surveyor, so that it could be seen (for instance) if he had a ring on his hand.
But this ‘concentration’ was very tiring and might become exhausting.
[...] Consequently it was only undertaken when information was urgently desired, and chance (aided by other information maybe) enabled the surveyor to pick out items (significant for him and his immediate concern) from the welter of the Stone’s visions. _ibid.

This analysis of minutiae required a significant amount of luck as well for the viewer to see through the haze and to be able to select the smaller points.

Answer 2

As for the Palantiri there was no one else to dominate. The Osgiliath stone had been sitting in the Anduin since the Kin-Strife (Gondor’s civil war). Two of the Arnor stones were sitting in the ocean, lost when Arvedui died in his shipwreck. The last one in Arnor only looked to the West and was used by the Elves of Lindon.

The Orthanc and Minas Tirith stones were the only ones Sauron could possibly communicate with. Not using them was the only way to impact what Sauron could do with his. No one was barging into Barad-dur to take it from him.

On the other hand, destroy the Ring, you permanently defeat Sauron who at that point was not going to be defeated any other way.