My DS is 9 and currently waiting for full assessment for ADHD; he was originally referred for a neurodevelopmental assessment into ASD and ADHD when he was 6, and diagnosed with ASD at age 7. At the time, because I didn't understand the subtleties of ASD presentation/development, I truly thought he'd be diagnosed with inattentive ADHD, or a mix of that and hyperactive. There's such a huge crossover in symptoms with ASD.
At 6, my son sounds very like yours, he had (and has) his moments of hyperactivity, that relentless, driven by a dynamo, energy, but certainly not always, and not really at school (that is, not really in the classroom, he would in the playground but that wouldn't be seen as deviating too far from the typical 6 year old). What was (and is) much more apparent was how he struggled with school work. Teachers would say he was in his own world. He was in intervention classes for maths and English, but even there the teacher had to have DS sitting by her so she could, as she said, "bring him back". He's seem to be not listening but quite often on questioning him, he could answer. But not always. He was and is working at below expected levels despite being of at least average intelligence. Home learning was what made me go and talk to his teacher, when he was in Y2 and start the ball rolling for assessment. It was always a terrible struggle to get him to do anything, but the one thing that made me realise that there was something more was after they'd had an internet safety day at school and they were asked to make a poster or leaflet, anything they wanted about something they'd learnt. Not only could he not do this, but he couldn't remember even having the special day. There was something about how he told me this that made me realise he wasn't bullshitting me, he truly could not remember. He also could never tell me what hed had for lunch, or what a story was about after we;d read it. At 6, he had started getting into Minecraft and Roblox and would play for hours, with hyper focus. He wouldn't listen to reason, would persist on a line or an argument (and he was argumentative), frustration tolerance was very low, he'd have to be repeatedly redirected back to tasks that were not of his own choosing (he really needed one to one at school to keep him on task). Behaviour wise, again, he'd be relentless and wouldn't stop doing something he'd been asked not to do until I made it impossible, eg, if we were on the bus and he wouldn't stop a certain behaviour, we'd get off the bus and walk, despite warnings up the wazoo (we no longer get buses, I pay for a train season ticket and it's worth every penny).
When he was diagnosed with ASD and not ADHD, I didn't immediately pursue the ADHD diagnosis as I felt school was offering him what they could anyway (classroom/teaching techniques, plus the intervention groups he was in because of his low attainment) and having the extra diagnosis probably wouldn't make a lot of difference in terms of what they could offer. But none of that stuff has made a significant difference to his attention/focus/concentration, whereas he's improved somewhat in terms of stuff that's specifically addressed the social skills side of his ASD, and consequently he's still behind academically. And as expectations are higher in terms of classroom and personal organisation, since he's in Y5, going into Y6 in Sept, and secondary school is very much on the horizon, I have pushed it. The gap between him and his peers is not getting smaller, and also of significance, he is starting to notice and become distressed whereas before he was oblivious/unaware/didn't seem bothered. In the end he was seen by an ed psych (through school, they got her in because of behaviour problems, mostly impulsivity) and she and I talked at length and after her assessment she referred him herself to CAMHS with her report as evidence/the basis.
I hope he gets diagnosed because I want to trial meds. He's got too much potential that's being lost. If anything will help I'll try it.