(1) The hosts of Men and Elves had no hope of winning.
Unfortunately, the movie fails to capture this important point: the enemy they had defeated in the plains of Pelennor was but a small fraction of Sauron's full might.
‘My lords,’ said Gandalf, ‘listen to the words of the Steward of Gondor before he died: You may triumph on the fields of the Pelennor for a day, but against the Power that has now arisen there is no victory...Hardly has our strength sufficed to beat off the first great assault. The next will be greater. This war then is without final hope, as Denethor perceived. Victory cannot be achieved by arms, whether you sit here to endure siege after siege, or march out to be overwhelmed beyond the River.
(2) Sauron didn't know about the Ringbearer at Mount Doom until it was too late
That we should wish to cast him down and have no one in his place is not a thought that occurs to his mind. That we should try to destroy the Ring itself has not yet entered into his darkest dream.
Sauron becomes aware of this plan only moments before it is completed.
And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him, and his Eye piercing all shadows looked across the plain to the door that he had made; and the magnitude of his own folly was revealed to him in a blinding flash, and all the devices of his enemies were at last laid bare. Then his wrath blazed in consuming flame, but his fear rose like a vast black smoke to choke him. For he knew his deadly peril and the thread upon which his doom now hung.
From all his policies and webs of fear and treachery, from all his stratagems and wars his mind shook free; and throughout his realm a tremor ran, his slaves quailed, and his armies halted, and his captains suddenly steerless, bereft of will, wavered and despaired. For they were forgotten. The whole mind and purpose of the Power that wielded them was now bent with overwhelming force upon the Mountain. At his summons, wheeling with a rending cry, in a last desperate race there flew, faster than the winds, the Nazgûl the Ringwraiths, and with a storm of wings they hurtled southwards to Mount Doom.
(3) It's not his style
He will not come save only to triumph over me when all is won. He uses others as his weapons. So do all great lords, if they are wise, Master Halfling.
In the Silmarillion, Sauron is Morgoth's chief lieutenant, yet he principally relies on deception and illusion, rather than participating personally in battles (unlike, say, Gothmog).
Recall that Sauron was once personally defeated, even while he wielded the One Ring. After that downfall, and now without his Ring of Power, there's even less motivation to take such a personal risk, particularly for a battle whose outcome is certain regardless.
We don't know how much strength Sauron had regained by this time. Probably quite a lot. In any case, there wasn't a reason for him to participate personally in these events.