Indeed. Furthermore, Aragorn helped fuel these beliefs. Not just with the attack on the black gate, either. He used the Palantir and revealed himself (and his sword), he arrived at the Pelennor fields with the royal standard unfurled, having routed the corsairs, and gathered the men of the coasts to his banner already. These actions had firmly placed Sauron's gaze on him. Depending on just how much was seen in the Palantir, Sauron may even have caught hints of what Aragrn did as Thorongil: setting back his plans by years, if not decades.
The march to the Black Gate was the (incredibly risky) strategy to keep the dark lord concentrating on the upstart heir of Isildur. Surely only somebody with the power of the ring behind them would be so bold, so foolish, as to assault the Morannon while heavily outnumbered. Who gives a stuff about spies on the Stairs of Cirith Ungol when this Ranger has given you the chance to extinguish the line of Elendil, at the same time as destroying the forces of Gondor and Rohan.
Your second paragraph is right on the money. Destroying the Ring was never something Sauron considered possible, because he would never have done it himself. Who would willingly surrender such power? Especially during a war where they were horrendously outnumbered, and in danger of a total collapse. The fact that the Men of the West then went on the offensive 'proved' that Aragorn had the Ring.
