I get it.
This might sound like a weird question, but how intelligent is he?
ADHD often (not always) comes with a higher than average level of intelligence - I think because ADHDers tend to have faster mental processing they can learn at a faster rate.
With DD (bearing in mind this was years before ADHD became a consideration), she was smart but was also way ahead of her peers in terms of emotional maturity.
This caused issues because she got frustrated with the behaviour of her peers, but also her intelligence meant that her practical knowledge and understanding outstripped her emotional understanding.
When she was about your sons age, we would ask her, in an age appropriate way, why she reacted the way she did, or what was going on in her head. She didn't always know, but when she could articulate it, it helped us to find strategies.
Couple of examples....
She would be happily playing with her sister and then all of a sudden have a huge tantrum, directed at DD1.
Initially we would tell her off make sure there were consequences etc. But when we were able to talk to her about it, she said the tantrum was because she was angry. She didn't know why something innocuous made her so angry, but it did.
We then made suggestions like if she felt herself getting angry she should take herself to her room. If her sister followed it was OK to tell her to leave her alone. In her room she could get her frustrations out by punching a pillow, or shouting if she needed to.
It didnt stop the bickering and falling out completely, but it certainly reduced it. She still punches her pillow or throws squishmellows at the wall now if she is frustrated and I've lost count of how many stress balls she's burst!
We also found that with things she couldn't help (like standing in front of the TV) teasing and humour had more of an impact than being told off. A lot of ADHDers hate being teased or laughed at because it taps into our RSD so it has more imapct than getting told off which is easily ignored because it feels illogical or has no emotional impact.
The phrase "Oi, DD, If your butt was made of glass...." still get's uttered regularly in this house, and we have got used to just ignoring the movement the rest of the time.
Your DS clearly needs the movement, so look at things like a balance board and set him challenges - "Let's see if you can stay on the balance board for 1 minute" then keep upping it.
While DD is watching TV and you are cooking dinner, set him challenges to do things around the house - make a treasure hunt or tell him to find five things that start with A etc (and then keep it going by getting him to put everything back). It helps to address competitiveness and the mental hyperactivity. Making treasure hunts for her sister also kept her occupied.
Craft is another one - we have a million unfinished projects lying around, but she would hyperfocus for hours on something crafty - had to be her idea though, usually involved her wanting to make something for someone else, and she had to be able to get up and move while she was doing it.
In terms of other practical solutions - can the TV go on the wall so that it can't be stood in front of or is there another room DD could watch TV in?
You don't need a diagnosis to employ ADHD strategies or to ask school to make adjustments. It is only needed for bigger, EHCP level adjustments and medication.
I would also say don't worry about it at this stage. While your DS may have ADHD, it sounds like its likely at the mid to lower end of the spectrum, in which case, if he and you can figure out strategies to manage the negative affects, there are positive sides which make it a super power 😉
Like another poster, I also thought a couple of your responses sounded a bit ADHD like and there is a genetic component to it, so if you haven't done so already, it might be worth you doing this screening tool: Adult ADHD Self Screening Tool - ADHD UK https://share.google/eTkVuTrSIo5oFRp5X