The thing is, for a significant proportion of children with ADHD (my eldest is one of them, my middle one whose ADHD is much more severe, is not) they can't fall asleep because their body literally doesn't produce the hormone that makes you feel sleepy at the right time. It's called melatonin, we should naturally start to produce it in the evening an hour or two before we need to sleep, but people with delayed sleep cycles produce it about 3-4 hours later instead - which could explain why your son is consistently falling asleep around 9-10pm when most children his age are tired and going to bed at about 7 or 8.
You can buy melatonin as a supplement in chemists in other countries, but in the UK it is only available on prescription. I understand people sometimes buy it online, I don't know what I would do honestly. I live somewhere it can be bought in the local equivalent of Boots so I spoke to my child's doctor, they okayed it and he's been taking it on and off since he was about 9. He's 17 now and still struggles to fall asleep at a normal time, in fact he will go fully nocturnal if he doesn't have to get up for school. When he was 5, I used to sit on his bed and keep reminding him not to move his body or talk/make noises with his mouth as these were the two things which would prevent him actually relaxing into sleep. I know that some children self-soothe with these things (DS3 does) but DS1 did not, it was like they were stimulating him to keep him awake. I did find though that if I told him to close his eyes, that was counterproductive because he would screw them up so tightly that his whole body was in tension, which would not help him fall asleep. But sat on his bed and just slowly counting to 3 if he started to move or make noise would help him go into more of a relaxed, sleepy state and then he would drop off within about 20 minutes, because he was tired. If you want to know how to absolutely optimise everything about sleep possible, there is a very nice book by Lyndsey Hookway called Still Awake, but a book can't solve a problem which is biological in origin.
One complication is that sleep deprivation in itself causes almost identical symptoms to ADHD. But if it's caused by sleep deprivation, and you can manage to fix the sleep deprivation, whether this is via behavioural methods (which don't work if they aren't sleepy because they aren't producing melatonin) or some kind of medical solution, the ADHD-like symptoms will go away. If it's actually ADHD then it doesn't matter how much sleep they get, they will have the same struggles anyway. However because ADHD in itself can be a cause of sleep issues (for various reasons not limited to late melatonin production) you can get stuck in a cycle where you can't fix the sleep. If this is the case then it makes sense just to proceed with the ADHD assessment.
I would speak to your GP. Go with the checklist. Explain about the sleep problems, see if they can suggest anything for that short term.