It is a grand, perhaps unsubstantiated claim to make, though the brilliant Celtic striker of the 1920s and 1930s was certainly prodigious. His total haul of 485 goals has never been bettered in British football.
What was less said about the great McGrory concerned his final, feeble years. Struck down by a form of dementia which dimmed his spirit, he finally died in 1982 aged 78, his mental powers having been notably damaged over a number of years. McGrory's trademark as a player - like many a great striker - had been his ferocious heading of a ball, the type, it was said, which usually gave a goalkeeper no chance.
This week the vexed issue of ex-footballers and early onset dementia has been aired again by the Jeff Astle case. The family of the former West Bromwich Albion and England striker of the 1960s are agitating in their 'Justice For Jeff' campaign for greater awareness of the brain damage suffered by footballers through their repeated heading of a ball - especially the old leather balls used up until the mid-1960s.
Astle, like McGrory, was famed for his heading. With his sudden death after collapsing in 2002 aged just 59, a coroner decreed that his passing had come through "industrial disease", referring to Astle's repeated heading throughout his playing career. In effect Astle's heading of a ball had amounted to prolonged, minor instances of repetitive cerebral blows.
Numerous studies, both in Europe and north America, have established a link between later cognitive impairment and the earlier playing careers of footballers, rugby players and American footballers. Previously known and written about mainly in the context of boxing, football is now coming under the microscope in terms of the damaging legacy it can leave.
Today there are a number of ex-pros in Scotland suffering from a degenerative brain disease, but who remain private on the matter. Not all cases, such as that of Frank Kopel, 64, who suffers from vascular dementia, can be directly linked to their careers. Increasingly, however, a number of neuropathologists are indeed establishing a link, as with Astle, in different cases.
Recently Dr Willie Stewart, a brain injuries specialist at the Southern General in Glasgow, established a connection between rugby and early dementia caused by repeated head-collisions in matches.
Statistically, Dr Stewart even suggested that two players involved each year in the Six Nations championship would go on to be afflicted by a related dementia.
In football the research is interested in the period from the mid-1950s to the early 1970s - in others words for ex-players who today are in their late 60s or 70s. That period witnessed the last days of the old, saturated leather ball - the type used by McGrory and others - before the introduction of the lighter, plastic balls, though these can still cause cerebral damage.
Ian Ure, now 74, played for Dundee, Arsenal and Manchester United in that period, and says he hears recurring stories of ex-pros suffering mental decline.
"I'm not surprised in the slightest that so many old footballers are suffering from some form of early dementia," says Ure. "When I think about my career in the late 1950s right through the 60s, you spent so much of your time heading a ball, either in training or in games.
"Funnily enough, I played against Jeff Astle many times, and he was a real aerial threat. That was a huge part of Jeff's game. You really had to watch him, he was a great header of the ball. But there was obviously a toll in all of that for him over all these years. It's very sad."
By coincidence, Ure as a lad also watched Billy McPhail, the former Clyde and Celtic striker, who suffered early onset dementia and claimed it was linked to his football career before his death in 2003.
"I saw Billy McPhail play a lot - he was a bit of a hero of mine," says Ure. "Billy could also really get up for the ball with his head. And that was with the old leather ball which, on rainy days, really did become very heavy.
"Nowadays I keep hearing of playing contemporaries of mine who are suffering from some form of early dementia. I always think it seems a very high percentage for guys who are actually not that old.
"I'm pretty sure the constant heading of a football, and the way it would make the brain shudder in the skull, is very much to do with it. It has to be. I've been lucky. Despite being a big centre-back I've had no issues at all on this score. But plenty others have."
Craig Brown, the former Scotland manager, who signed for Rangers in 1958 and then went on to play for Dundee in the 1960s, says it is no surprise that some ex-players have suffered a degree of brain damage, given the conditions in the old days.
"The weight of these old leather balls could be horrific in wet weather," says Brown. "The fact is, if it was coming at you out of the sky on a wet day, you didn't want to head it. You really felt it."
At Rangers as an 18-year-old, Brown says he was issued with boots with metal toe-caps, to protect the players' toes from the ball.
"The game was harder back then," says Brown. "It was much slower compared to today, granted, but back then it required a lot of courage. Heading those old balls, by God, the impact of it. You would shudder from the impact."
What Scotland probably requires, given the anecdotal evidence of early mental impairment in former rugby players and footballers, is a systematic study of the effects of their playing careers.