It is also true that his reluctance to see England's victory over Andorra this week as some sure-fire harbinger of success in next summer's World Cup was entirely justified.
"Andorra are possibly the worst side I have ever seen," said Greavsie.
Indeed they were wretched but then hardly more so than when England failed to score a first-half goal against them in their previous two encounters. Certainly it is embarrassing to remember the reaction of the then England coach Steve McClaren when Steven Gerrard eventually broke the first of those deadlocks. The coach declared that "Stevie G" had again announced himself a world-class player. Of course he hadn't, not then. He had merely got the better of a glorified pub team.
This brings us to the point which Greaves might have raised without automatically joining the cheerleaders who crowd the broadcasting booths ever more relentlessly these days. It brings us to Fabio Capello, one of whose strengths is that he is no more likely to claim bragging rights over Andorra than appear in the technical area dressed in crumpled trackies.
Capello merely contented himself with celebrating the fact that his team have acquired some basic self-belief and technical competence. He might not have put it quite like that but it was, demonstrably, the way he was feeling in the wake of the 6-0 win.
What even Greaves could have reasonably persuaded himself to say about Il Capo as he moved a little closer to a perfect World Cup qualifying record, is that the England cause has been immeasurably strengthened and rationalised under his command. Think, for example, about the way he has handled the David Beckham situation. It is nothing less than Solomonesque. Apart from seeing a generally poised England do what they should have done – which is to say score six goals – the remarkably high number of fans who braved the traffic chaos were also rewarded with a second-half exhibition by their favourite David Beckham.
In terms of the realities of top-class football it was a nice exercise in nostalgia but utterly meaningless. Capello would no more have played Beckham in a central midfield position against a serious football nation than he would someone wearing a Mickey Mouse suit.
But then what did that matter against the possibility of pleasing everyone: the Football Association, the fans, the sponsors? Like any half competent coach after two and a half years' work, Capello knows his best team – and he knows that Beckham, when the serious action starts, is not in it. He is a bit-part player, albeit no doubt one of considerable potential value towards the end of the game, and his starting place is on the bench.
Yet consider how skilfully Capello has both made that last point so clear and still managed to preserve Beckham's starry march to the status of England's most capped outfield player. After years of the gesture politics of Sven-Goran Eriksson and McClaren, some of us were not quite prepared for the subtlety of Capello's approach.
Yes, he admitted privately, he was a little surprised at the weight of public and media pressure for Beckham to win a 100th cap. Indeed it was a phenomenon that he would not expect to occur in Italy in a million years. However, his predecessor McClaren had made a stand against Beckham and, in acutely embarrassing circumstances, responded to the pressure to reinstate him.
No such embarrassment was courted by Capello. If the people wanted cake, give them cake, he seemed to say. If they wanted Beckham, and perhaps even more importantly the people upstairs wanted him for reasons not totally to do with what happens out on the field, well give them Beckham – but not, crucially, at the cost of seriously compromising the development of the team. Beckham could stay around, stockpile his caps, produce cameos that inevitably produced a few examples of the best of his old game. And the manager could get on with the job of making a new team and a new culture.
None of this guarantees England's ability to break into the competitive company of favourites Spain, Brazil and Argentina in South Africa next summer, but it does remind us of the one indisputable lesson of this week's Andorra experience. It was that in Capello, England have a coach who understands not only football but the wider world. He may be outgunned against the Spanish midfield of Xavi and Iniesta, he may have nothing to match Lionel Messi's responses to the urgings of Diego Maradona, but plainly he has the tough pragmatism to get the best from what he has got.
He has made believers of Wayne Rooney and Gerrard and Frank Lampard, who under Eriksson and McClaren suggested they needed formal introductions before a game-plan. We cannot imagine Capello going to a major tournament, as Eriksson did in the European Championships in Portugal in 2004, and then debating the shape of his midfield. Even more unlikely is the possibility that he would have gone to not one but two World Cups, as Eriksson did, with key players like Beckham and Michael Owen unable to guarantee their fitness.
Capello told Beckham that if he was to continue his pursuit of international caps, he had to maintain his fitness and play some serious football. Only heaven knows what he would have said to those England players in Istanbul who, on the eve of a vital European Championship qualifier, were murmuring about a strike to protest the FA decision to suspend Rio Ferdinand following his failure to take a drugs test. We can be sure it would not have been Eriksson's unequivocal statement that he would remain on the fence.
Capello may not win the World Cup. He might not even match Eriksson's routine quarter-final placing in major tournaments. But if there was any lingering doubt about his major achievement it was surely removed at Wembley this week. He has made England a grown-up football team, and if you catch Jimmy Greaves in one of his less acidic moments he might just admit as much.
Twenty20 has case to answer after Wright's swinging misses game's point
I would not wish to embroil my esteemed colleague, cricket correspondent Stephen Brenkley, in any of the fall-out that followed an assertion here earlier this week that Twenty20 cricket is a crude and shoddy perversion of so much that is the best of the game. However, those of us who are ready to accept a blindfold and a last cigarette before surrendering the belief that Twenty20 is indeed a grievous offence against arguably sport's most civilised game, will surely be excused if we plunder a little of his fine reportage this week when England were eviscerated by the South Africans.
Wrote Brenkley, "England chose to bat on a pitch which had been used earlier in the day and would have wished to set an innings of somewhere around 150. Ravi Bopara chopped a slower ball from Dale Steyn on to his stumps in the first over; in the second Luke Wright edged behind his THIRD [my capitals] optimistic drive of the over. Wright's rustic swinging has severe limitations."
As they say in court, "We rest our case, my Lord."
A speech to bring hope to romantics
On the day that Cristiano Ronaldo celebrated in the company of Paris Hilton his £80m move to Madrid, another hugely gifted young footballer made a speech that suddenly made the English game seem a whole lot less denuded.
"I see my future here," he said. "The boss knows I really want to stay. I hope I can be with this club for many years and lift trophies they have never lifted before.
"I am really looking forward to it because I am sure the future is bright here."
You could only hope that it was a statement grounded more firmly in truth than Ronaldo's recent jokey split-screen interview with himself, when he assured United fans that he was going to stay.
The author of the loyalty report is Cesc Fabregas. It is not only Arsenal supporters who will pray he means what he says.
If he does, it means that the idea of football as something more than an exercise in grab-all self-indulgence will have been considerably enhanced.
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