Such thoughts are surely there, but another issue is prominent too: he and his advisers are obsessed with how to restore people's trust in politicians. It was a top priority long before David Blunkett's messy fall from grace put the spotlight on Mr Blair's failure to live up to his promise that Labour would be "whiter than white" after the sleaze-ridden Tory years.
Mr Blunkett's departure illustrated that the list of measures to clean up politics in Mr Brown's bottom drawer is needed urgently. The system for ensuring that ministers stay on the straight and narrow is a mess. Incredibly, it is less strict than for MPs, who have an independent watchdog, the Parliamentary Standards Commissioner.
Yet the ministerial code of conduct is not policed by a watchdog with teeth. The Cabinet Secretary can interpret it, but the final arbiter is the Prime Minister.
So when Mr Blunkett admitted to The Independent on Sunday he had breached the code by refusing to get clearance for the outside jobs he took after resigning as Home Secretary, Mr Blair could judge that he should not be punished. True, Mr Blunkett did resign in the end. But if Mr Blair had decided that keeping him would cause less damage than losing him, he would still be clinging to his cabinet post.
Some ministers claim the system worked because Sir Gus O'Donnell, the Cabinet Secretary, confirmed Mr Blunkett had broken the rules and Sir Alistair Graham, chairman of the anti-sleaze Committee on Standards in Public Life, criticised him in media interviews. But they could not punish Mr Blunkett. Nor could the Advisory Committee on Business Appointments, which he failed to consult. Its rules say that former ministers must approach it before taking outside jobs, but are free to ignore advice.
"The system is farcical," one senior Whitehall figure admitted yesterday. "There are too many overlapping bodies and they have no teeth. They didn't force Blunkett out. He only went because of pressure from the media and the Tories."
Downing Street claims the ministerial code is under "active consideration". But there is not much sign of activity. The idea of appointing an independent regulator has been in Mr Blair's in-tray on several occasions but he has always shuffled it to the bottom. Similarly, he has repeatedly shelved proposals for a Civil Service Act to safeguard the independence of civil servants and limit the remit of political advisers.
Mr Brown has not yet decided to tidy up the rules for ministers and civil servants. But an obvious move for him as incoming Prime Minister would be to surrender control of the ministerial code to Sir Alistair's independent committee.
That would be consistent with other measures in Mr Brown's mind to rebuild trust by strengthening Parliament's powers. He has spoken of giving MPs the final say on approving military action, a signal that he would not repeat Mr Blair's breach of trust over the Iraq war.
The Chancellor also wants to reform the House of Lords, which can only mean one thing - most, if not all - peers being elected. He has spoken of the need to "reinvigorate local democracy".
As his critics see him as a control freak, he would need to do something dramatic. Perhaps he will. After all, he handed over the power to set interest rates to the Bank of England.
Maybe Prime Minister Brown would go the whole hog by introducing something akin to a written constitution. That could also resolve potential conflicts between the European Convention on Human Rights and Britain's anti-terrorist laws.
Critics would say there would not be many votes in all this. But there might be if people saw it as a genuine attempt to clean up politics.
And there is another reason to try. As one Blairite minister put it: "If David Cameron becomes Tory leader, his pitch will be 'I am Blair without the lies and spin'. That could be very dangerous for us."
Some people will doubt whether Mr Brown is the right man to sweep the stables clean since, after his long spell as Chancellor, he will be seen as part of the political establishment. Voters will be wary of such promises because they think Mr Blair failed to keep his.
It won't be easy to rebuild public trust in our politicians. It may be too late. But at least Mr Brown is going to try.