I agree a school wouldn't push for it unless there were clear signs, but at the same time, they are generally only aware of signs that directly impact schooling. My son is now 8 and awaiting diagnosis (but we are in little doubt as both I and my sister have it) I had an inkling before the school, I first started thinking it was possible when he was 3.5, but until 6, school would say...possible but he is a July baby and a boy...so could just be normal hyper young boy. His year 2 teacher then brought it up without mention from me and school have put things in place for him ever since, though that same teacher told me I would need to push for them to do anything about it, because he wasn't struggling academically.
Things to note that relate to your comments, he is also a very bright boy, though one of the youngest in the year he still sits around middle of the class for results, he loves to learn as long as he is interested, that is the part people often don't realise about ADHD, you can focus great for long periods of time if the thing you are doing or learning is interesting to you, but when it's boring you have no hope, he was dry at night at instantly once he was dry in the day, so that's not a given link. Though he was a hyper little man, he is also anxious and his biggest struggle is actually transitions and change...task switching, get very anxious over new places, new teachers, and his confident bubbly self only comes out once he knows where he stands and the rules etc.
However we did see clear signs at home too, the biggest being emotional regulation. He has BIG feelings, and frustration levels go from 0-60 pretty fast! Impulse control also, he would do something and then almost instantly feel bad, the action comes before the brain has thought it through.
The wait lists are long, so don't think there is any harm starting the process, you may see more signs as he gets older or school might see less signs, so you can always not follow through with it later. You don't say if he is in Yr 1 or yr 2, but loads of kids struggle moving from reception to yr 1 as there is such a drop in movement and play based learning in the UK, so it might just be a case of him being a mix of clever so easily bored (there is a big gap between oldest and youngest in the yr at that age) and not getting enough time to move around.
It could also be that he does have ADHD But cos he is relaxed at home and you scaffold him well it isn't that noticeable
I would be surprised if it was ADHD and you didn't see any emotional regulation issues at home though.
Side note. I am starting to suspect my little girl (almost 7) will also be diagnosed as some point, but I doubt school will see it as she is also clever and she is well behaved and works well at school (she is also a perfectionist, takes any mistakes or perceived critism to heart, big ADHD red flags in girls) So it presents differently for all. I was late diagnosed and no one believed I had it but me, and yet hear I am, diagnosed and getting worse as life gets more busy and oestrogens levels plummet! 😅😆