The supermarket sits right where that angry, rickety old stadium once stood.
"I always stop and visualise it in my mind," says McMillan, a former great of Airdrieonians and Rangers.
"The supermarket is right where the old pitch was and I'll sometimes pause and think, 'right here, I?¯remember, I?¯missed a sitter that day.' I?¯can more or less exactly visualise the spots."
McMillan, the famed "Wee Prime Minister" of Ibrox, was 10 years an Airdrie player and still lives in the town, but it was his six years at Rangers between 1958 and 1964 which elevated him to greater glory. Not least because this much-loved, beautifully balanced footballer was always different.
"Even in my six years at Rangers I?¯was never full-time," he says. "I?¯was always part-time. Oddly, I?¯trained at Airdrie three nights a week but, when I?¯signed for Rangers, for some strange reason, I?¯then trained only two nights a week, even though I?¯was playing in Europe.
"I?¯was a quantity surveyor. I?¯had a good job and career away from football. I?¯realised that, at 27, which I?¯was when I?¯went to Rangers, all I?¯needed was to get an injury and I?¯could be finished in the game. So I?¯wanted to keep my full-time job.
"It was a dilemma when I?¯signed for Rangers but the club said they were happy for me to remain part-time. I?¯never saw any of my team-mates until we met up on a Saturday for the game. I'd train at night."
McMillan offers a fascinating insight into football in those far-off days of the 1950s and 1960s. The Rangers manager who signed him was Scot Symon, a quiet, painfully distant figure whose onerous task had been to follow Bill Struth, but who had huge success before leaving Rangers in unhappy circumstances after 13 years.
"In those days managers didn't tell you what to do; the players just did their own thing," says McMillan. "The manager picked the team but that was all. We basically played a 4-2-4 at Rangers. Jim Baxter and I?¯were in the middle of the park. We didn't plan it; it was just the way it worked out. The job of Baxter and I?¯was to get the ball to our strikers.
"Scot Symon was a very successful Rangers manager, but he just left everything to the players. It was totally up to us to do what we did on the park. You heard nothing, no instruction, coming from the dugout. In those days the manager wasn't even there, Mr Symon was up in the directors' box. The players were left to their own devices."
I'd heard this sort of thing before from older players - the near-passivity of football managers of old - but it was still weird to hear it again from McMillan.
"What Scot Symon had was a good managerial sense: he knew how to fill the positions with the right players. The only thing Symon did was, at the end of the game, if he passed you and you'd played well, he might say: 'That was well done.'"
But Symon did not exude warmth.
"He was a quiet man. I?¯actually don't think he liked praise or anything like that."
What a team Rangers had back then in the early 1960s, with McMillan and Baxter the inspirations. The club reached a European Cup semi-final in 1960 and a European Cup-Winners' Cup final in 1961, as well as accumulating numerous Scottish honours.
In its midst, there were McMillan and Baxter: the former a clean-spoken, church-going, erudite man, the latter a feckless sybarite who just wanted to have a good time. McMillan on Baxter is fascinating.
"Jim Baxter was a silly man in many ways. Had it not been for his behaviour, he could have played for an awful lot longer; he was such an inspirational player. He didn't need to think what he was doing on the pitch - it just came naturally. He was absolutely brilliant - he went by men as if they weren't there.
"I?¯used to go in to Ibrox on a Saturday morning, only to hear what Jim had been up to on the Friday night. If he hadn't played for Rangers he'd have been locked up some nights. But he had a good streak in him as well. He could be kind and generous. The shame of it is, but for his lifestyle, he could have played for far longer."
Older Rangers fans remember Baxter as a world-class midfielder, but the creative, flighty McMillan also drew great praise, despite his own modesty. "I?¯had a lot of minuses as a footballer," he says. "I?¯wasn't good in the air. I?¯couldn't head the ball. I?¯wasn't fast. I?¯couldn't really tackle. So I?¯had quite a few minuses about me. On the plus side, I?¯could control the ball quite well. And I?¯was pretty accurate with my passing. I?¯could also 'take men on' as they used to say. I?¯had a bit of trickery about me, though nothing like Jimmy Johnstone's, I?¯might add. But I?¯could get by the man."
It was often said of McMillan that "you never once heard him swear".
In other words, he and Baxter never went out clubbing together. McMillan has been steeped in the church all his life, and remains so. "We were a church family," he says. "I?¯started in Sunday School and went all the way through. It's a shame: the church is on a downward trend. My church is amalgamating with another church. It's sad to see."
After his playing career, McMillan managed Airdrie for six years, and took the club to the 1975 Scottish Cup final, where they lost to Celtic.
I?¯needed to ask him one last thing about the old Broomfield, the angriest pews I?¯ever came upon in football. "Oh yes, and it was worse being a manager," said McMillan, laughing at the memory of that notoriously abusive old main stand. "You had to walk from the old pavilion all the way round in front of that stand, and if you lost, you had to walk all the way back.
"It was well known back then for its abuse. Even in my day as a player, it was known for its reputation. Nobody was spared. If you responded you were just asking for more trouble. It was no bed of roses being the Airdrie manager.
"That abuse was part and parcel of being the manager there."