Obviously in the real world there aren't 3 levers to adjust: there are probably thousands or tens of thousands. Apart from the lobbying, politicians may themselves desire this, because it would allow them to lower those 3 very-public levers to claim they lowered taxes, while at the end of the day maintaining the same level of spending because they just offset those losses over thousands of other levers. After all, things have to get paid for.
Then at least they can tell the public: "Hey, I lowered tax rates." And when the public's wallet doesn't feel the savings, the politician still wins votes because as far as the public can see, they lowered those tax rate levers. Compare that to a politician under the 3-lever system. Either they lowered taxes or they didn't.
What I'm suggesting is that a complex tax system allows politicians to take the heat off themselves when the public demands lowered taxes, while still maintaining the amount of money the government takes in to cover the budget.