Obviously George Bush didn’t govern via Cabinet meetings; they were just photo ops when he had them. But what about Clinton, Bush I, Reagan, Carter, Ford? I actually have no idea.
In brief, Eisenhower presided over a modest revival in the idea of “cabinet government” that was aided by the fact that he didn’t have an especially ambitious governing agenda. But then under Kennedy, LBJ, and Nixon the cabinet’s influence waned. Gerald Ford came into office determined to repudiate the Watergate legacy which was seen as linked to an all-powerful White House. But Ford came to feel that this was leading to chaos, and over time the White House was strengthened (under Don Rumsfeld and Dick Cheney) in order to keep the cabinet on a tighter leash. Jimmy Carter went through essentially the same cycle. Ronald Reagan learned the lesson of Ford and Carter era failures and didn’t really try, neither did George H.W. Bush. Clinton started out with neither a strong cabinet nor a strong Chief of Staff, but always relied heavily on White House staff and moved toward appointing a strong chief of staff.
Now none of this is to say that some cabinet officers can’t be extremely influential. Certainly Robert Rubin was a key policymaker as Secretary of the Treasury. But the nature of modern government is that the Secretary of the Treasury was hugely influential because he was Robert Rubin and not vice versa. Rubin was hugely influential as head of the National Economic Council (i.e., as part of the White House staff) for the same reason he was influential later — because Bill Clinton thought relying on his advice was a good idea. On top of that, one of the big problems with cabinet government in the modern era is that in some respects the cabinet secretaries are too important. The federal government is way bigger than it was in the heyday of cabinet government and the cabinet secretaries need to run their departments. That’s hard work. It means that the can’t spend all their time together in meetings at the White House working out big-picture policy issues. But many topics — especially where you want to shift policy significantly — require interagency coordination. So recent presidents have found two models to deal with this. You can either have chaos, with different departments going in different directions, or else you can have a strong White House staff to make sure things stay on track. Nobody’s found a feasible way to get the secretaries to do the necessary coordination themselves.