He was already publicly branded a traitor during the Council of Elrond. It wouldn't have done him much good if he had tried to personally take the Ring from the Nine Walkers. "What if I got captured? What will they do to me?"
Saruman had no need to leave Orthanc at all; he had servants, slaves, lackeys to do his bidding.
Denethor (and Sauron) share this viewpoint:
Denethor laughed bitterly. 'Nay, not yet, Master Peregrin! 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. Or why should I sit here in my tower and think, and watch, and wait, spending even my sons? For I can still wield a brand.'
The Lord of the Rings: The Return of the King, The Siege of Gondor
He was playing the "better safe than sorry" tactic. Instead of going with his Orcs to battle, he stayed at home. After all, Orthanc is impenetrable. Even the Ents couldn't damage it.
To the centre all the roads ran between their chains. There stood a tower of marvellous shape. It was fashioned by the builders of old, who smoothed the Ring of Isengard, and yet it seemed a thing not made by the craft of Men, but riven from the bones of the earth in the ancient torment of the hills.
A peak and isle of rock it was, black and gleaming hard: four mighty piers of many-sided stone were welded into one, but near the summit they opened into gaping horns. their pinnacles sharp as the points of spears, keen-edged as knives.
The Lord of the Rings: The Two Towers, The Road to Isengard
The Uruk-Hai of Saruman were unquestionably loyal to him
'That can't be helped,' said another. 'But why not kill them quick, kill them now? They're a cursed nuisance, and we're in a hurry. Evening's coming on, and we ought to get a move on.'
'Orders,' said a third voice in a deep growl. 'Kill all but not the Halfings; they are to be brought back alive as quickly as possible. That's my orders.'
'Is that all you know? Why don't we search them and find out? We might find something that we could use ourselves.'
'That is a very interesting remark,' sneered a voice, softer than the others but more evil. 'I may have to report that. The prisoners are not to be searched or plundered: those are my orders.'
'And mine too,' said the deep voice. 'Alive and as captured; no spoiling. That's my orders.'
The Lord of the Rings: The Two Towers, The Uruk-Hai
Even when presented the choice to search Merry and Pippin, the Uruk-Hai of Isengard didn't give in to temptation.
'Aye, we must stick together,' growled Ugluk. 'I don't trust you little swine. You've no guts outside your own sties. But for us you'd all have run away. We are the fighting Uruk-hai! We slew the great warrior. We took the prisoners. We are the servants of Saruman the Wise, the White Hand: the Hand that gives us man's-flesh to eat. We came out of Isengard, and led you here, and we shall lead you back by the way we choose. I am Ugluk. I have spoken.'
Indeed, Saruman didn't trust his Orcs to the hilt, but he knew that they were perfectly capable of following orders (which they did), as long as they didn't know anything about the Ring.