My comment may be lost on you, as I feel we are looking at life very differently, but I'll try, and perhaps you can think about it. I would be interested if others have experiences that are similar to mine.

Let me give some background first...

I grew up like most kids, going to school, but also perhaps unusually for the time (20-30 years ago), spent nearly all my free time playing in the fields and woods and riding bikes with my small group of friends. Honestly, it was the highpoint of my life so far.

From about age 6, I was obsessed with "living in the wild" and survival skills. I read books about it, and taught myself skills such as plant identification and friction fire-lighting. I did exceptionally well at school, mostly through natural ability, but also a fear of getting told off, which meant I always did as I was told. It turns out that if you are naturally intelligent and stick to the task, you do well. Who knew?

By about about 13, I began losing focus. At first I still did well, but in the end, I left school with a mixture of As and Bs. It was made clear to me that for someone of my calibre, this should be regarded as a failure.

Looking back, knowing what I now know about mental health, it was clear to me that going to secondary school had been emotionally devastating for me -- I was overwhelmed by the pettiness and the bullying -- and was increasingly depressed as time went on. On reflection, the tone of pettiness and bullying was actually set by the teachers, and in fact, the school system itself.

Skipping forward, I became an arborist (tree-surgeon) when I was 20. Even as a previously active person, I discovered a depth of "animality" that was unknown to me. I became profoundly tuned to my strength and power. It was liberating. All this time, I found it difficult to socialise, as I found other people so boring, not because of who they were, but because of what they did, or rather, didn't do. Once everyone got to age 16, they took up drinking, which is just an elaborate form of spending your time sitting down, which I found intolerable. When they weren't drinking, they would maybe go to the cinema, or drive around in their cars, all activities a million miles away from my outdoor interests.

It wasn't until I was 27 years old and suffered a knee injury, that I learned how to be comfortable sitting down. I didn't really like it, but I got used to it. Since then, I have spent plenty of time sitting down, the internet has taken over all our lives, and I have developed chronic sleep problems. I've never felt worse. I never did go and live off the land; it turns out the UK isn't really set up for that kind of thing.

So that's my experience of life. I don't know what yours is. It's probably quite different. From my perspective, I have zero difficulty understanding where children are coming from. Think about it. You're born an unknowing blob, unable to even focus you're eyes on things, but you're growing, and have many built-in urges. As you learn, naturally, to walk, you begin to explore and learn about the world. In the first years (and in fact, you're whole life) your outlook on the world just keeps getting bigger. You need adults to watch your back, but you want to get out there on your own too. How? Walking, running, climbing, engaging those animal instincts that I found so intoxicating in my twenties after years of education.

But then you reach an age (5) where your world starts to get smaller; you're sent to school by people who you thought cared for you and a stranger makes you do things that you don't want to do. A year later, things change and a different stranger bosses you around. This pattern repeats every year. If you ask mum or dad why you're doing this, it's said that it's for your own good.

Maybe the early years aren't so bad, but for my own part, even as as kid with no formal training in exercise physiology, as I got older, I had a visceral sense that I wasn't getting enough exercise, and would strenuously devote all my free time after doing homework to being outside, even in the dark winter nights. I was lucky, as I had friends who would do the same, but still, it was boring at times.

Restaurants are designed for adults, with refined sensibilities in cuisine and conversation. These sensibilities are usually lacking in children who just want to eat and get back to playing.

Your son was hitting himself over the head. Sure, why not? After all "[you] sternly reminded him to sit down, [leaving him] apparently rueful that he's disappointed [you] again". I don't know you, so won't judge your parenting, but you might want to judge yourself, by noting how many times a day you sternly remind him of things. If it's frequent, then what's in it for him? It's one thing to be corrected now and again, but if every day is one correction after another, then what does he have to look forward to? Clearly not family time at the restaurant. You might be surprised just how upset this all makes him. If people are upset and feel they have no control, then they tend to act erratically and "uncontrollably".

There is debate in the other comments about whether ADHD is over-diagnosed and people "just need more exercise". Your kid hasn't even been assessed, let alone diagnosed, so while other people in the thread may be talking about ADHD, we are not. We are talking about your perception of your child's behaviour. Hopefully, I may have given you a different perception.