This is a key plot point: Sauron never even thinks of the possibility that the Ringbearer will try to destroy the Ring. He cannot imagine somebody voluntarily giving up that much power. From the book:
Gandalf: For he is very wise, and weighs all things to a nicety in the
scales of his malice. But the only measure that he knows is desire,
desire for power, and so he judges all hearts. Into his heart thought
will not enter than any will refuse it, that having the Ring we may
seek to destroy it. If we seek this, we shall put him out of his
After the Fellowship split, Gandalf and Aragorn work hard to maintain this deception. They want Sauron to believe that Aragorn has the Ring.
When Pippin uses the Palantir at Isengard, Sauron sees a hobbit for a few seconds. He thinks that Saruman has captured the Ringbearer.
After Isengard is destroyed, Aragorn uses the Palantir to taunt Sauron. From that moment, Sauron believes that Aragorn has the Ring.
Aragorn then does exactly what he knows Sauron expects him to do if he had the Ring: raise an army and march to the Black Gate to make a frontal assault on Mordor. This was a terrible risk, taken to keep Sauron's attention and forces focused on Aragorn.
And note one more point: Sauron was right. Nobody could voluntarily destroy the Ring. When the time came to throw the Ring into the fire, Frodo could not do it. If Sméagol had not been there, Frodo would indeed have succumbed to the Ring's temptation and tried to wield its power.
In a letter to a reader, Tolkien described what would have happened next. Sauron would have immediately sent the eight surviving Nazgul.
I think they would have shown 'servility'. They would have greeted
Frodo as 'Lord'. With fair speeches they would have induced him to
leave the Sammath Naur [Mount Doom] - for instance 'to look upon his
new kingdom, and behold afar with his new sight the abode of power
that he must now claim and turn to his own purposes'. Once outside the
chamber while he was gazing some of them would have destroyed the
entrance. [...] In any case a confrontation of Frodo and Sauron would
soon have taken place, if the Ring was intact. Its result was
inevitable. Frodo would have been utterly overthrown: crushed to dust,
or preserved in torment as a gibbering slave. Sauron would not have
feared the Ring! It was his own and under his will.
Tolkien Letters 246