Even Blair friends and defenders like columnist Martin Kettle are quick to acknowledge that "the perception of Blair's relationship with George W. Bush has been catastrophic for him, absolutely catastrophic."
The legacy of this special relationship may well overshadow all that Blair achieved during his 10-year stint as prime minister.
So what are the unqualified successes of the Blair decade?
The very fact of an entire decade spent in power, say some in the Blair camp.
As Hoggart pointed out rather puckishly, "It's very revealing that when anyone says to him, 'What are your greatest achievements?' he always says, 'Getting elected three times.' So, why did he want to get elected? To have achievements. What's the achievement? Getting reelected!"
Well, setting a record as the longest-serving Labour prime minister is no small feat.
Nevertheless, there are some other achievements to be reckoned with.
Chief among them is the establishment of a power-sharing government in Northern Ireland, a fact not lost on Blair. His resignation announcement comes only two days after this historic agreement went into effect, long enough to savor the pleasure but short enough to ensure that people remember him for it, in the midst of all the discussion about legacy.
Even critics like Hoggart concede that "he carried that through with an enormous amount of personal skill, energy and imagination. I don't think anyone can take that away from him. Northern Ireland is now a decent, prosperous place to live."
Then there are the other defining moments of the Blair leadership era.
His response to the death of Princess Diana -- summed up in his words, "she was the people's princess" -- saw him "anticipate and capture the public mood" with total accuracy, according to the columnist David Aaronovitch.
Aaronovitch told ABC News that "in many ways, at that moment, Tony Blair helped save the British royal family," whose own reaction to Diana's death was felt by many to be inadequate, as depicted in last year's Oscar-winning film, "The Queen."
Then there was the astonishing reversal of fortunes in July 2005, when London found itself at the top of the world one day -- voted to host the 2012 Olympic Games -- and in the depths of despair the next, when the city was hit by a series of bomb blasts, all carried out by homegrown suicide bombers.
Aaronovitch reflected that "under those circumstances there was every danger that the country would be polarized." But the government's reaction, he concluded, "is regarded by both his friends and his enemies as a substantial moment for him. In other words, he did the right things in the right order at the right speed."
"In many ways," Aaronovitch said, "he is better in a crisis than he is when there isn't one."
Except, of course, when that crisis is the Iraq War and its aftermath, described by Parris as "a pretty sorry and confused, stumbling spectacle."
In the end though, it may not be the Iraq War that was the undoing of Blair. It may just be the fact that he was far too presidential to be prime minister.
Jenkins told ABC News that Blair "loves the presidency. He loves Washington, the limousines, the power, the glory. It's an open secret at Downing Street [his official residence] that he longed to be a president."
Unfortunately, that was not to be, despite the fact that in recent months, Blair's popularity in the United States eclipsed his standing at home.