Having lived a rich and eventful life, he had found it impossible - sifting through a multitude of professional triumphs and personal joys - to pick out his greatest day. But there was no hesitation in identifying his worst. Today marks its 43rd anniversary.
Few will need reminded of what January 2, 1971 represents, or that 66 people lost their lives, crushed to death as they made their way from a football match. Those who attended that infamous Old Firm derby will never forget. For the rest of us, images of horribly twisted guide rails are vivid enough hints of the horror that unfolded.
Fallon merely witnessed the aftermath, and that was appalling enough. Celtic's assistant manager watched as, throughout Ibrox - in dressing rooms, on tables and on the turf itself - the dead and dying were laid out alongside the badly injured. He had shepherded the players out of the stadium to make room for the casualties, and alongside Jock Stein, Neilly Mochan, Bob Rooney and the Rangers backroom staff, remained behind to help wherever he could.
What he remembered - he was speaking just a few weeks before his death, tears welling in his eyes - were the young faces. The dead had, after all, included five school-friends from the same Fife village, and 31 teenagers in total. The youngest victim was just nine. Hence the certainty with which Fallon reflected on the Ibrox Disaster over four decades later.
"That was the worst day of my life," he declared unequivocally. "Seeing all those kids lying there and having to carry their bodies, I'll never forget it. For a young boy to go to watch his team and never come home, it should never have been allowed to happen. It was just heart-breaking. I shed more than a few tears, and I'm not ashamed to admit that I still get emotional thinking about it.
"What happened that day was more important than all the football in the world. At a time like that, you'd gladly forget about trophies and winning Old Firm matches if those people could be back with their families. I think everyone who had been there that day would have gone home and just thanked God their own family was safe. And you never forget about it. How could you? I can still see those poor boys now. It breaks my heart just thinking about it."
At Fallon's side amid the chaos had been an old playing adversary and long-time friend, Willie Waddell. The then Rangers manager likened the initial scene to Belsen, saying that the tangled corpses reminded him immediately of the images from the Nazi concentration camp. "My God, it was hellish," Waddell recalled. "My own training staff and the Celtic training staff were working at the job of resuscitation, and we were all trying everything possible to bring breath back to those crushed limbs."
As someone qualified in first aid, having trained as a lifeguard during his days as a champion swimmer in Sligo, Fallon was especially busy. And his efforts were not in vain. Indeed, while researching the Irishman's biography, I learned of a youngster - seemingly left for dead - who owed his life to Celtic's assistant manager. The boy in question had stopped breathing, and with paramedics overwhelmed and focusing on those with the best chance of survival, his frantic father had pleaded with Stein's deputy to do what he could.
"I was trained to give the kiss of life, so naturally I tried my best to save him," Fallon recalled. "After a while, we thought we could see some movement. So we kept at it and eventually, thank God, the lad started breathing again. He survived, I know that. That was one good story to come out of that day; I just wish there had been more like it. All I know is that it wasn't for the lack of trying, and it wasn't just me. Bob Rooney [Celtic's trainer] and the Rangers physio - anyone who knew what they were doing, in fact - were going round, looking to help whoever they could."
Though instinct had kicked in those crucial early hours, Fallon returned home to the full, devastating realisation of what he had witnessed. "That was a dreadful night," remembered his wife, Myra. "Sean just sat there, sobbing. We couldn't get anything out of him. All he kept saying was that these poor boys had been lying there without a mark on them."
Billy McNeill wrote in his autobiography that it was with "the look of haunted men" that Fallon and Stein returned to work the following week. Both attended several of the ensuing funerals, while Fallon led the offertory procession at a special St Andrew's Cathedral service attended by players, staff and supporters from both sides. A few days later, he watched approvingly from the touchline as Celtic Park observed a sombre and flawless minute's silence.
Amid the desolation, hope did flicker briefly that the disaster would prove a watershed not only in stadium safety, but in relations between Glasgow's warring football tribes. "This terrible tragedy must help to curb the bigotry and bitterness of Old Firm matches," was the wish expressed by Stein on the front page of the following week's Celtic View. It would be a forlorn hope. The resumption of hostilities, and the return to business as usual, was swift.
Yet Stein's description of the bitterness in that same article as "sordid and little" was a view long held by the manager and his deputy. "We had no time for all that sectarian stuff," said Fallon. "All the division and hatred; it was a load of nonsense. The players, certainly when Jock and I were playing, wanted nothing at all to do with it. We played hard against each other, but we were great friends. I often wished the supporters could have seen that.
"We would go out socialising, and I got on very well with the likes of George Young - big Corky we called him - and Willie Thornton. Deedle [Waddell] would be there too; he and I were great pals right up until he died. He was one of the best players I ever came up against. He was big for a winger and, naturally, I'd try to cut him down to size a bit.
"We got on very well. He even proposed me for Pollok Golf Club, which was very tough to get into at the time. He told me: 'I'll get you in, Sean. But no kicking me when we're out on the course.' I told him I was making no promises."
This Rangers legend was someone of whom Fallon spoke often. "Deedle felt the same way about his team as I did about mine," he would say admiringly. To the Irishman, this devotion to a rival club made Waddell a kindred spirit rather than an enemy.
Fallon never lost sight of the fact that they were united by a great deal more than that which divided them. And he never forgot the day they were united in grief.