Kumar Sangakkara is freezing at the end of an abandoned day of county cricket in Sussex. Wearing a huge borrowed jumper from the visitors' dressing room occupied by Durham, for whom he played two matches while preparing for Sri Lanka's one-day and Test series against England, the great batsman still smiles through the rainy cold. The longer he talks, and the deeper he thinks, the more his eloquent intelligence and personal warmth crackles like a small fire of hope.

After an hour in his company it is hard to avoid a nagging contrast. England's team is full of bright and engaging men but, lately, it has become absurdly difficult to talk to them. Access to England's best cricketers has been so controlled that even a request, in the recent past, to interview a junior player has had to be tied to "a sponsorship opportunity". Such encounters often feel like a carefully managed exercise to curb honesty and personality.

Sangakkara, one of the giants of modern cricket, is different. He has played 547 international matches for Sri Lanka and hit 35 centuries in 122 Tests. His Test average, 58.07, exceeds the imposing figures posted by Rahul Dravid, Ricky Ponting, Sachin Tendulkar and Jacques Kallis – his now retired contemporaries. Yet a true measure of Sangakkara's quality is heard when he addresses issues of conflict and disaster, tolerance and diversity, tragedy and victory, with a lightly authoritative touch that transcends the need for a media officer or a corporate plug.

"It's been an amazing privilege to play the game," the 36-year-old says. "I have a lot to be grateful for as so much comes down to personal relationships – even when signing an autograph. It's more a case of taking time to say: 'Hello‚ how are you? Thank you for coming.' The child might value that personal interaction more than an autograph one day. We undervalue that, but it's what we all need. More human contact, more human interaction – exchanging different ideas and philosophies should be our priority."

Sangakkara believes in cricket's capacity to heal divided communities. Last month, while leading Sri Lanka to victory over India in the T20 World Cup final with a half-century, he was reminded of the power sport can wield when allowed to flourish freely. His undefeated 52 was a minor classic that assumes even more gravitas when it is remembered that he and Sri Lanka had lost their four previous World Cup finals – and that Sangakkara struggled for runs earlier in the tournament. But when it mattered most, he found serenity and steel.

"I might've looked calm," he says, "but on the inside it was more turbulent. Mahela [Jayawardene] and I were batting together when he got out. I thought: 'Well, I'm still here. Let's see if I can get us over the line,' and I had the privilege of doing it. Three runs before we got it I thought we'd won. I celebrated a bit, but 'no, too soon, too soon.' Then Thisara [Perera] hit that six."

Sangakkara was hoisted aloft by his team-mates and paraded around the ground before, back in the dressing room, he felt at a loss. "You feel so happy, but so tired. What should you do? Do you keep celebrating? Or just think about what happened? Call your family? Share the news, share the joy?

"I actually sat there and absorbed this feeling of satisfaction, and gratefulness to everyone. I'm a Buddhist. And we have Christians, Hindus and Muslims in this team. You have that feeling of being humbled – that you need help from the Divine, friends, family, supporters. You need all that to say I've been part of a World Cup-winning team.

"But no one expected the reception we got back home. There was a 40km unbroken chain of people and it was very moving – because we saw every single Sri Lankan of whatever religion or creed. It was one big family welcoming home 15 of their children who had achieved something they really valued. It touched me deeply because cricket has always been more than a sport and that continues – even more so today in the context of post-war society and building inter-community relationships."

In 2011, during his coruscating yet poetic Cowdrey lecture at Lord's, Sangakkara exposed the corruption at the heart of Sri Lanka's cricket administration. Has progress been made since then? "I think huge efforts are being made to ensure history does not repeat itself. There's full recognition of what it will take to rebuild Sri Lanka after we had suffered so long. So the healing continues and, in cricket, great efforts are being made to ensure we drive towards a new future."

Few cricket teams carry the same acute social consciousness and Sangakkara pauses when asked if this is a consequence of a civil war that, starting in 1983, lasted almost 26 years? "The team also represented something so unique, so loved and longed for – the diversity of people, ethnicity and religion and how we manage to understand and value each other and perform so well. That instilled a sense of responsibility in us.

"The tsunami [in December 2004] was a big turning point. Seeing the devastation, alongside Murali [Muttiah Muralitharan], Mahela and other cricketers when we went up the coast with supplies, changed us. We spoke to people and you saw the desperation and tragedy in their eyes. We met a man who had lost all his children and his wife. And he stood by himself. It was terrible – and so the cricket team transformed itself."

Sangakkara sometimes visits the Terrence Higgins Trust in London, on his own, to learn how best to eliminate the stigma against people afflicted with HIV. The most meaningful work, as always, is done without fanfare. "We don't have to be in the newspapers handing out things. It's not just about giving money or posing for photos – it's giving your time and being able to play a positive part in someone else's life. It's easy to give money and walk away. So we don't do charity as a choreographed team exercise. But man of the match and series awards are put into a kitty and that money goes to people who need urgent medical treatment."

Sangakkara also describes the lessons learned after the Sri Lankan team were fired on by 12 gunmen in Lahore in 2009. He was one of six injured players, sustaining shrapnel wounds to his thigh. "It just brought home what other people endure every day. I faced just three minutes [of gunfire] but it gives you a new perspective on the resilience of our people to come back from tragedy. Everyone on that bus, within a month, was back playing, and playing well.

"I was stopped later at a checkpoint in Sri Lanka and a soldier said: 'Hello‚ how are you?' I saw he was so upset and shocked. I said: 'But you go through this every day.' He said: 'No, it's my job. You are a hero. This can't happen to you.' This was a man who gets a nominal salary to fight for his country and every time he stops a vehicle in Colombo his life is in danger. It was so humbling for him to say we were heroes. But that's heroism – to place someone else's life above yours and say: 'My job is to protect you.'"

The Lahore attack formed a compelling segment of his Spirit of Cricket lecture. Sangakkara fused the personal and political with evocative impact as he also remembered how his father, an attorney, opened his house to 25 Tamils forced into hiding in 1983. Did he have any doubts talking so intimately, and politically, to the conservative MCC?

"I was terrified at the start, looking at the room. I had sent though an advance copy that evening as I finished writing it two hours before I gave the speech. I walked into Lord's and CMJ [Christopher Martin-Jenkins] was there. I said: 'What did you think?' He said: 'I think it's going to be very good – and very interesting.' He had this little laugh and I thought: 'OK – maybe it'll go well.'

"When they stood up at the end I thought they were doing it out of politeness. It was a standing ovation but they said it was only the second time, after Desmond Tutu, it happened. So maybe they did appreciate it. It also caused a big impact at home. The little negativity was dwarfed by the hugely positive response. It was hugely satisfying that it resonated with different generations."

Looking ahead to this summer, the Twenty20 match is on Tuesday with the 50-over series starting on Thursday. Sangakkara has retired from the T20s and will concentrate on one-day and Test cricket – and play until the 2015 [50-over] World Cup. "Every game is important as I won't be doing this too much longer. Whether it's club, county or international cricket I'm going to play as hard as I can and enjoy it as much as I can [in his final innings for Durham against Sussex last week he scored 159]."

Test cricket matters most to him and Sangakkara smiles when reminded how he once gauged a batsman's quality. He said, in 2007, that anyone who scored 20 Test centuries could classify himself as a "great" batsman – but 30 hundreds elevate a player to an exalted level. What does he think now, after his 35 centuries? "It gives satisfaction but it makes you look for new landmarks – not for recognition's sake but because you know you can do better. You can't sit back and think: 'Oh, I've had a great career‚ thank you very much.' You can always do more – which means scoring more than 35."

England have a new coach, Peter Moores, who had previously held the position, while his assistant Paul Farbrace coached Sri Lanka to their World Cup win last month . Farbrace is an Englishman and so Sangakkara is understanding and effusive. "Fabby's a top man. He responds very well to people and he helps them respond very well to him. That's worth more than a million hours of analysis. It's unfortunate things happened in the way they did. We would've loved to have had him with us – but Fabby's a great addition to England."

Did Sangakkara see much of England's recent humiliation? "Everyone follows the Ashes because it's the only iconic Test series – which is sad because, for the longevity and sustenance of Test cricket, you need more series of that nature. Australia played magnificent cricket and England were out-bowled, out-batted, out-fielded.

"It might have surprised England after they had dominated Australia for a few years. But the Aussies have a very good side with Mitchell Johnson, Brad Haddin and newer players like Steven Smith, Chris Rogers and David Warner. They had tough times and came back strongly. Sometimes when things don't go your way tempers can be lost. But you need difference of opinion and characters. You sometimes need an argument to get honesty, responsibility and accountability."

Sangakkara has given those three virtues, alongside intelligence and generosity of spirit, to Sri Lanka and world cricket the past 14 years. And few other Test cricketers would say, of Oscar Wilde, "the artistry has always stuck with me", or relate an amusing story of reading Charles Dickens to his children. "I thought I'd start with David Copperfield," Sangakkara says, "but they were howling by page two. I stopped – even if my wife couldn't stop laughing."

When the day comes, perhaps next year, and Sangakkara finally stops playing internationally, cricket will be a much poorer place. Until then we should savour a man who personifies all that is best about a game that, to him, is always far more than a sport.