After almost 20 years as a professional player, including with Tottenham Hotspur and Norwich City, he completed a master's degree in psychology before working in performance psychology, including with the Crystal Palace academy. That combination of playing experience and academic study has left him particularly interested in one of football's biggest remaining questions: how do you actually measure what is happening inside a player's mind?
For McVeigh, the contrast with the physical side of the game is striking.
"I think it's really challenging because a lot of it is subjective," he tells Football Presse. "Because if you're really going to start trying to equate and correlate the psychological state with the physiological state, it's really easy to measure the physiological states because the technology, the wearable devices and stuff that we have now is so, so advanced.
"I've even got one, I've got an Oura Ring on now, which is measuring my stress, my heart rate, my HRV, my sleep pattern. I also have a Garmin watch on, which does the exact same thing. Heart rate measures, all the different bands that I'm training at every single time I do any exercise. So the physiological side has evolved so much."
McVeigh has seen that transformation first-hand. When he joined Tottenham's youth system in 1994, the infrastructure around young players was almost unrecognisable compared with the modern game.
"When I started at Spurs in 1994, the staff for the Spurs youth team was a coach and a kit man. You didn't even have a physio because the physio, we used the first-team physio.
"I'll fast-forward that to 20-odd years later, 25 years later, and I'm at Crystal Palace working as the sports psychologist for the academy. The academy staff — this is 10 years ago — we had two coaches, two physios, a masseur, a doctor, a sports scientist, an analyst, a data scientist, a strength and conditioning. It was just — and that's for the academy. What do you think the first team had?"
Tottenham's own development reflected the wider transformation.
"Two years later, after I started in 1996, Spurs brought in the first sports scientist for the entire club, professionals and 20 youth-team players, one sports scientist. But that was the start, and then they just went like that."
The explosion of sports science has made the physical condition of a player increasingly measurable. The same cannot be said for their psychological state.
"If you go to the psychological state, the psychological state is only assessed through observations, which is really hard to understand what's going on in someone's head. Or asking the person. And then of course you have to take in, are they lying? Are they telling the truth? Are they giving me a wee bit of it? It is so difficult to try and find that out.
"So it's almost like the science part for the physiological side, they've got it nailed. The psychological aspect is way more of an art. And of course, it then comes down to the practitioner of how they actually get that out of the individual."
That difficulty becomes even more significant when the player is expected to discuss their mental state with somebody working inside the club.
McVeigh returned to professional football for a year, working with a Championship club, and found players could be reluctant to reveal what was really happening.
"If you have a psychologist or someone who's working like a mental performance coach working with the players, the goal or the ideal scenario is that a player can go in and speak to that coach or psychologist and almost like offload, take things off their chest, share things in a way that I'm struggling with this, this is hard, I've done this. It's a really, really intimate relationship.
"The problem that I've seen, and it was actually only 12 months ago I was still working in professional football, just because when I went back in kind of under the radar with a Championship club just to do stuff for a year, was that players were so guarded that they didn't want to share anything of value.
"Because they thought it was going to go back to the manager. Not because they didn't trust me, just because they thought it's going to go back to the manager. And then you think, as a player, why am I giving my weaknesses, my insecurities, and all the things that I have wrong with me that I'm kind of going to hide when I walk into the training ground every day?"
That creates a problem for the very people who are supposed to help.
"You don't want to tell it to anybody, whether it's a performance coach, a physio, or a psychologist. You might end up putting it back to the manager because then they think the manager's going to go, 'Oh, if you've got this issue, then we're not going to play you.'
"So you see, it's like that catch-22. I think what's starting to happen is, if — and again, I can only surmise and hypothesize this — I think that players, if they are interested in working with a performance coach, mental performance coach, they'll do it outside the club. And they'll just do it secretly so no one knows what they're doing because that external coach is not going to have any way to relay it back to the coaching team."
There is another problem. Even when researchers can demonstrate a relationship between psychological and physical factors, McVeigh is wary of reducing the human body to a simple equation.
Asked whether a player's psychological state can influence injury recovery, he points to research suggesting that positive approaches, meditation and other techniques can help, while stressing that the evidence cannot be reduced to a straightforward cause-and-effect formula.
"It's really, really difficult. And I don't know where you're getting your questions from, but these are really challenging because you're at that intersection of how do you actually understand cause and effect?
"This is critical thinking, which is difficult to say that X plus Y equals Z because X plus Y does equal Z, when all of these factors are taken into the equation. But actually, as soon as you go to the next day, already X plus Y is different.
"So again, you're almost — you can't replicate this cause and effect. And this is where, again, when I did my master's in psychology, it just completely opened up this entire world of we can say things with certainty, but we can't say it with full 100%, this is how it is."
That uncertainty does not mean McVeigh believes psychological training is ineffective. Quite the opposite. His own playing career provides an example of why he believes mental rehearsal deserves to be taken seriously, even if its precise effect is difficult to prove.
As a teenager at Tottenham, his father sent him a Paul McKenna visualisation tape called *Fantastic Football*. McVeigh repeatedly imagined receiving the ball on the left, cutting inside and curling it into the far top corner.
"I used to listen to it every morning, every night, and practice this visualization of essentially rehearsing peak-performance me. And the one scenario I had was getting the ball on the left-hand side of the pitch, cutting in on the left-hand side and curling it in the far top corner.
"For some reason, I don't know why, but that was the scenario that I used to visualize every single day. And I did that for weeks and months and years all the way through my career."
Then came the curious part.
"The one goal that I scored more than any other through my entire career was getting the ball on the left-hand side, cutting in, and curling it in the far top corner.
"And my question then to the audience is, because I did that visualization, is that why I scored those goals? As the majority of my goals was the exact same scenario that I used to visualize."
McVeigh refuses to claim that visualisation directly caused those goals.
"I can't say it didn't have any impact. I can't say it was just down to the visualization. It's somewhere in the middle.
"The question that I would probably maybe ask people to be curious about is, do you think that's a coincidence? Do you think it's a coincidence that the one scenario I used to visualize over and over, day, weeks, months, and years, season after season, and at the end of my 20-year career, the one goal I scored more than any other in my career was the one scenario I used to visualize?
"I don't think that's a coincidence. But if I have to try and sell that to a young player, do visualization because it's going to help your career, it's a very, very difficult argument to make."
For McVeigh, that may be the fundamental challenge facing psychology in football: not necessarily proving that the mind matters, but finding ways to understand and measure exactly how much it matters.
"The physiological side has just advanced. The psychological aspect is way more of an art."
And in a sport increasingly obsessed with measuring everything a player does, what happens inside their head may remain the hardest thing of all to quantify.