What about allowing you to choose a few (3, maybe 4) members of a cabinet to implement the decisions you make, and to offer suggestions of their own? The choices for each position which be drawn from 2 or 3 contenders for each job. Their willingness to push various ideas of yours, and the suggestions they make, would be colored by their background, which you would know in advance–but you wouldn’t see how well they’d implement/drag out your policies, except in the results. There’s also the possibility of running a sleaze risk, if a cabinet minister’s background includes hints of problems (tax evasion, several divorces, etc).
The idea is that a cabinet minister could actually accelerate changes that are supposed to occur over time, if you one favorable to a particular process you’re trying to implement. So while you could try running the government entirely by yourself, if you want clean air, or tax reform, you’d want somebody to push it all through at perhaps 120% of the rate you’d otherwise get. Of course, they might also slow down some of your other policies to 50% of their rate, if they disapproved of your direction. And they might even quit; you could see how pleased or displeased they were with your decision making. Quitting would allow you to choose somebody else, but also bring its own fallout from the groups who thought of a particular cabinet minister as being a friend in government.