I will just say that both of my kids who have been diagnosed with ADHD (DS1 at age 10, DS2 at age 6) I had (nursery) teachers and pediatricians insist there was nothing to investigate at age 4, and for DS1 that persisted up to about age 8/9 - his teacher was even surprised when I gave her a form to fill in as part of the assessment, and then proceeded to rate him as worse than we had on many of the measures. Nursery staff, teachers and non-specialists like HVs and GPs often have a picture of a particular "type" of ND and this only represents a small proportion of cases. I know MN is probably a bit too much the other way inclined, but I think what MNers tend to pick up on is a parent who is struggling with behaviour in a way which is outside the norm. And I realise that you posted this on a day when you were feeling particularly down about it but I also read your post this way and personally, I think it would be helpful for doctors/teachers/etc to recognise a sort of "high intensity" category of DC who may or may not later get a specific diagnosis, but in order to recognise that some children ARE hard work at this age and parents/families/the child/the school might need some support. I actually think this would be hugely beneficial because this kind of behaviour is so very difficult to respond to both kindly and effectively and you can easily get into a negative reinforcing cycle (like I described with the hypothetical not-ignoring-actually-deaf person).
With DS2 the question of assessment for ADHD only came up when he was about 5.5 and I also reckon it came up quicker because of the family history. In general, medical and educational professionals are very, very cautious to suggest things like this especially at an age where it's difficult to assess (and 4 is too young really to assess for ADHD, and ASD where there aren't significant communication difficulties).
I sense maybe a bit of fear in the response like if she is ADHD, then the behaviour won't be something she'll grow out of, it will always be like that. And I just wanted to add that DS1 definitely grew out of a lot of the more intense "hyperactive" type behaviour, despite his ADHD diagnosis. But I'll stop focusing on ADHD specifically because I can see it's not helpful.
OTOH I disagree with the sense that high energy or high intensity behaviour can't be reduced or managed because it can - it just takes quite specific approaches and knowledge. Sensory seeking behaviours seem useful to learn about because quite frequently if you have a high energy child people will say things like oh you need to get a trampoline/put them in drama lessons/get them doing gymnastics. And if you're lucky and hit on the right combination of activity with whatever sensory input they are seeking, then this will work to at least take it down a notch, and that's why people insist on it so strongly. (Proprioceptive input, which is a sort of deep body pressure is the most common one and is common to a lot of these activities people often suggest). But it's not always as simple as getting them loads of physical activity and tiring them out. If you don't hit on the right sensory input and/or if the behaviour is more a sign of dysregulation and the dysregulation is coming from places other than a high sensory threshold, then they simply won't seem to tire from these things and it might even make them worse because then they are tired and possibly frustrated as well if they struggle with coordination or taking instruction from others or whatever. And then you're left wondering WTF am I supposed to do, this child does not stop and nothing satiates them.
In my observation, the shouty/argumentative/needing to be right/first/etc behaviour tends to stem from difficulties with emotional regulation and social skills - they struggle to predict or understand the behaviour and nonverbal communication of others, or fail to notice it, so can't predict what people will do and because of this they find it quite an alarming experience to be around others because any time anybody tries to interact with them, they frequently interpret this as some kind of attack. Then because they have reacted defensively, this usually causes the other person to be surprised and think that the child is being mean/defiant and so children will stand their ground and be defensive back, adults will adopt a more firm response in order to correct that behaviour, either of which the child then feels is a further attack and both cements their original "rightness" in reacting defensively as well as causing their own defensive state to escalate as well. The emotional dysregulation causes the emotions to escalate more quickly so every reaction seems out of proportion, like a toddler who is constantly overtired.
When DS2 was anywhere between 3-5 and DS3 was anywhere between 1-3 himself I had to do quite a lot of sitting with DS2 (sometimes holding him back!) and talking deliberately calmly and slowly to sort of explain what I saw DS3 doing and what he could expect. For example if DS3 had a toy and DS2 wanted it, it didn't work to do the usual explaining about sharing/turns. I had to hold DS2 on my lap and explain that while DS3 has the toy now, he probably won't be playing with it for very long, we would wait for 5 minutes and he probably will have finished with it by then. (DS2 at this point DESPERATE to get down and snatch the toy because he doesn't believe me and/or can't see past "I want toy now"). By about 4 min 30 sec he usually had finished and DS2 would be surprised. But after a few times of doing this he slowly started to trust that I was probably right and we had fewer fights over toys. Not saying it never happened but it did reduce. Likewise DS2 very into building train tracks, which DS3 would happily dismantle. DS2 became fearful about toddler sibling coming near train track, even though I would defend it or move DS3 away and this persisted long after DS3 had left the random dismantling phase. I used to sit with them and point out to DS2 what DS3 was actually doing and that it was not the thing he was afraid of, what to say to get DS3 to stop/change, and that even if he was in a dismantling mood, we could also rebuild the bits of track he got to before I managed to interrupt him.
It is still quite exhausting doing this, but it tends to help much more than lots of reactive correction after they've already done the thing you don't want ie no you can't hit him, no you can't shove him out the way etc and constant breaking up fights. Because they frequently feel as though their original behaviour was justified/the only thing they possibly could have done, telling them off for it and applying consequences makes no difference, you will get the behaviour again. Trying to follow it back to the root and getting in before it starts and explaining from their own perspective tends to be more effective although it is much more intensive upfront.