WHEN the president finally pulled the rug from under John Sununu, it was George Bush's longtime buddy, Secretary of State James Baker, whose voice was most insistent and decisive in bringing about this action.Mr. Baker had talked President Bush into putting Mr. Sununu in the chief-of-staff position in the first place. Bush didn't resist. He owed so much to the former New Hampshire governor, who had helped turn the floundering Bush campaign around in '88. But he might have offered him a different position had not Baker told him that Sununu would be just the right person to be the president's right-hand man. Neither the president nor others supporting this appointment, however, ever intended this to become the "Bush and Sununu" show - with Bush as genial leader and Sununu as the "tough-guy." Initially, the president and those around him liked Sununu. They were amused by his droll sense of humor. And they were impressed by Sununu's brilliance (which he never hid) and his efficiency. Why shouldn't a governor - experienced in running a government and in the politics of getting elected - make a good chief of staff? And why shouldn't he be superb in helping Bush shape an agenda and put it through Congress? This rationale led to Sununu being tapped for a position where he might well have succeeded, but didn't. Sununu earned his dismissal - and that is what it was. Bush, known for his loyalty to colleagues, finally became convinced that Sununu's abrasive ways had made him a political liability. Son George Bush Jr. took soundings and reported a thumbs down on Sununu. But it was the secretary of state who provided the final push when he told Bush that Sununu was hurting the president's reelection prospects and simply had to go. Sununu was a victim of his own arrogance. He irritated those he dealt with - in Congress, in the media, within the White House, and elsewhere in the administration. Having said that, I want to add this: At first, Sununu was particularly effective in helping shape the Bush presidency. A lot of criticism heaped on Sununu was unfair. Because of his personality he drew attacks really intended for the president. Sununu was an easy target. Bush, like Reagan, is easy to like; Sununu is the kind of person people love to bash. And so we come to Sam Skinner, former transportation secretary, who now moves into Sununu's hot chair. Mr. Skinner wanted the job. That tells a lot about him and his self-confidence. He knows that if he falters he can damage his career. He knows that other recent chiefs of staff - Bob Haldeman, Don Regan, and now John Sununu - found the experience quite painful. Baker, again, gave the final OK on Skinner. He'd been hoping that Dick Cheney, Jerry Ford's chief of staff, would be willing to move from Defense back into a job he had once performed so well. But when Mr. Cheney said, in effect, ve done that job," Baker enthusiastically endorsed Skinner. Skinner is a skilled politician, learning the trade under the tutelage of former Illinois Gov. James R. Thompson. He has no rough edges, or at least they haven't shown up yet. Bush has set up a '92 campaign team "headed" by Robert Mosbacher, who is stepping down as secretary of commerce. Mr. Mosbacher will mainly be a fund-raiser. It will be Skinner, once Governor Thompson's campaign manager, who really will be running the Bush bid for reelection. Bush himself, of course, will be making the big calls. Skinner and pollster Bob Teeter will shape campaign strategy and recommend policy positions to the president. Skinner will actually run the Republican national organization. Thus, Skinner will quickly have a chance to prove his worth by charting a course for the president in the coming campaign. The final test will be the election. If Skinner can play a major role in reelecting the president, his selection by Bush as chief of staff will look like an excellent choice.