Well, I’ll raise a different angle:
I tend to the view that Blair is singing from exactly the same hymn sheet as all his immediate predecessors, including the likes of Harold Wilson and Mad Dog Thatcher. It is, of course, something fairly significant that could see such varied creatures of the night singing harmoniously, have them risk their own personal popularity and be such pro-active enthusiasts - as they have all been, in their in own way and in the quite seriously shifting circumstances of the past 30-50 years.
Let me illustrate by the example of 9/11:
From, say, about 10.00am East Coast/3.00pm London and for (at least) 24 hours, the US Government was not answering the phone. Period. No response, no plan, silence.
Blair (at the TUC Conference) knocks up a speech supporting the US and delivers it in place of the planned address. Then it’s a quick conference huddle with the inner circle before a never-ending day and night on the phone – including being the first (possibly only) international phone call Bush would have taken.
Result: Not influencing but, rather, informing the views and policies fermenting within the US Administration at that time, easing EU/European concerns over the nature of the likely US response and, once reassured by Blair that Bush wasn’t losing the plot, a coordinated European response to the attack based on that knowledge that Georgie wasn’t going to go banana’s (at least immediately). That response (from Europe) included the usual diplomatic niceties and general condemnation but it also allowed Blair to present a too tempting fait accompli to Georgie: an agreed framework supported by all the potential allies i.e. NATO. That was, given the shortness of time and difficulty in grasping the Very Large Indeed Picture, a beautiful piece of work. IMHO.
Georgie didn’t necessarily know it, but he wanted NATO for the (later) “world is with us” stuff and Europe wanted NATO because it acts (in this scenario) like a brake on any potential excesses (from a first term president who has but a passing acquaintance with passport control). And it keeps the communication/information lines open. The world settled down again understanding some kind of real-world framework existed in response to an event that was unthinkable and (diplomatically) unplanned for.
Yet no other outsider would have got to the Bush inner circle and informed opinions there, gauged the tone of that opinion (for the worried Europeans) and put together a planned response that satisfied all parties so very, very quickly at a time when no one had the first idea of what was going to unfold, or even about how to control events. It was a most worrying time. No question.
For that short period, Blair’s was the only hand on the tiller and, more importantly, the only possible hand on the tiller.
And that’s the UK’s game: The only way it can influence events on the world stage is to punch far above its weight. Aside from economic and military considerations (because they’re unrelated to the OP), the way the UK does that diplomatically is to have the American ear (at some superficial cost) and to use that to its advantage elsewhere. Example: Basic tenet for just about every diplomat at the UN – If you want the Americans to consider something, anything, you have to run it past the Brits first.
If there’s only one point of interest in this policy, I tend to think it’s how the role the UK has played has changed (increased) in the post-Cold War period. Now, we have one super power with one close ally, and that in spite of a more unified Europe (which, in other circumstances, might be thought to be challenging the diplomatic role of the UK). Without the USSR, the UK’s role has morphed into something quite different.
Thus, the playing field itself and the rules have changed dramatically yet the UK remains 10 points clear with three games to go: Hugely dynamic policy, (generally) extremely well judged and wonderfully pragmatic. And not a little sophisticated. I think, whether one agrees with the general thrust or not, the policy - in practice - is a real piece of work. Hence the support of every PM, even in the face of considerable personal cost.