It's probably worth remembering that John Major, too, long enjoyed a reputation as a nice guy. "People look at me and they think I'm a pretty straight sort of guy, and I am," Tony Blair famously said at the height of the Bernie Ecclestone/Formula One/Cash for Influence affair early in the life of the New Labour government.
Its leader sometimes affects a slightly bewildered air, giving the impression of a man taken aback by some of the nastier things which seep out from the Labour body politic. "I just don't know where this stuff comes from" is a classic Blairite response to questioning about, say, backstage briefings against Gordon Brown.
But you don't need an overdose of cynicism to know Mr Blair didn't get where he is just by being nice. This, as one commentator put it yesterday, is a leader who "contrived to abandon 50 years of ideological baggage without even a crocodile tear". There is a hard, ruthless side to the man who almost single-handedly made Labour re-electable. And it was on show during Tuesday's big speech.
For while Mr Blair was setting new missionary heights with an agenda for a radical century ahead, he might also have prompted two members of his cabinet to wonder if they had, after all, won their very public battles to stay in their posts.
Mr Tony, as he's sometimes called, might make light of it - the "froth" of politics. But some in the Mowlam camp had wondered aloud whether she could repeat last year's conference triumph, when that great, spontaneous Movation almost upstaged the leader's speech. And, more to the point, whether she would be allowed to. Those who trade in the sort of gossip to which the leader pays no heed fancied the Northern Ireland Secretary had done herself no favours last time around. She had even registered in one poll as more popular than Mr Blair - a position most people with an ounce of political intelligence could instantly recognise as dangerous indeed. So there were confident predictions that Bournemouth '99 would see no repeat performance of Mo mania.
And if all politics was personal, that would be understandable enough. Ministers, even good and valued ones, need to be kept in their place.
It is certainly no part of the leader's function to encourage the development of ideas above their station. (We may be sure Number 10 has duly noted Ken Livingstone's assertion that some on the left would back Mo for leader, should the great man ever fall under the Clapham omnibus).
Yet all is not personal, and sometimes mere pride has to take a definite back seat. The spin doctors might have been determined to curtail any expression of Mo madness, but it might have been thought that events of recent months would actually have decreed a political need for some show of the Prime Minister's continuing affection.
All summer long, unionists have been demanding Dr Mowlam's head. For all the subsequent denials, it seems clear that Dr Mowlam - like Health Secretary Frank Dobson - eventually got it into her head that Number 10's faceless men were responsible for talking up Peter Mandelson's chances of succeeding her in the Stormont hot seat. Different people would offer different theories about Mr Blair's subsequently aborted reshuffle. But one persistent view was that Dr Mowlam's reluctance to go, and an inability to find her a suitable alternative, were a major factor in staying Mr Blair's hand.
In her only pre-conference interview, Dr Mowlam appeared to indicate that she was still minded to stay. Most definitely her reasons for wanting to hang on back in July had not yet been satisfied.
And she would have known, as she spoke, that the next wave of speculation was imminent, ahead of an enforced reshuffle following George Robertson's departure for NATO in the middle of next month.
She would also know that, post-Patten - with an intensive debate on the future of policing ahead and with Mr David Trimble contemplating what might be his final risk for peace - he [Mr Trimble] and his followers are more impatient than ever to see the back of her.
Mr Blair could have quashed any fresh doubt about her survival on Tuesday afternoon, as he could about Mr Dobson's.
The Prime Minister spoke at length, and with feeling, about the NHS without any pat on the back for the Health Secretary he apparently thinks should vacate his post to stop Mr Livingstone becoming Mayor of London. And while his reference to Northern Ireland was considerably shorter (a mere 12 lines) again, there was no word of praise for his Secretary of State.
On Monday afternoon Dr Mowlam - last year's conference darling - had less than a starring role, kicking off a wide-ranging debate to a fairly empty hall, while most of the representatives were still queuing up at the security barriers after lunch.
Now here she was, seated beside Cherie, the spotlight assuredly on her, conspicuously denied any share of reflected glory. What a difference a year makes.
Maybe, of course, there was no message here. Perhaps the fine political calculation was that the current difficulties in the peace process would have rendered any hint of self-congratulation inappropriate. Perhaps. But, if so, those news managers will again have shown themselves less adept than their reputation suggests.
And if, to their intense irritation, the speculation revives again, they will only have themselves to blame.