You don't need a diagnosis to look up coping strategies and how to use and teach them at home.
Schools should keep an eye on the kids getting a lot of detentions - including how they react to the detentions. I think schools should be strict but watchful and any punishment should actively include looking at how to cope if you are struggling in this area.
If smartly done, punishment shouldn't just be punishment but an opportunity to support and identify problems - I do think this is where schools are possibly failing and should do more. That's where the mental health aspect is failing not with the strict discipline itself.
The issue is bigger with kids who don't have that home support - but that's not an ADHD issue alone. It's something else. And should be identified and flagged at that. Parents not wanting to take responsibility to parent is different to parents actively neglecting though. A parent who is spending all their time on the phone to school arguing the toss defending little Johnny and not actually working to improve the situation is demonstrating they don't want to take any responsibility. It's a partial school thing but it also has to be a partial home thing. (Which is why it's a world of difference from the completely disinterested parents and should be treated like that). I actually think a fair amount of strict style schooling is about parental management as much pupil management.
FWIW DS's diagnosis from start to finish took from near the start of yr3 and we just got it during the summer before yr5. During that time primary school have been good with him despite a lack of diagnosis and treated him giving the benefit of the doubt. He's now waiting being forwarded on to other support services.
I don't think this one should just be on schools to deal with. A lot of issues are home related so you have to deal with it at home. There's loads of stuff online about this.
Both DH and I have strong indications of also being ADHD. For me it's proved a problem in my adult life and it's been identified as a possibility recently since it's come up with DS and understanding how it presents in girls. I've had burn out issues and other diagnosis. I don't want DS to have the same problems I do and DH doesn't want DS to have the behavioural issues that affected his education and he felt unsupported by his parents about that he did.
My issue is a diagnosis of ADHD seems to be interpreted as a limitation on life which then restricts everything in life. We need to get away from the idea that ADHD means a shrug and a 'oh well they can't be responsible' attitude. That's far from the case. And it's unhelpful in the long run to kids. I know the areas I struggle with as an adult and even having an understanding of ADHD has helped enormously.