I would push now because of the huge waiting times tbh. Still apply strategies and learn about it yourself, but it’s a long process being assessed.
Equally, lots of people will claim someone with HFA is simply gifted, but, if they do have HFA and it’s missed, it can be invalidating because those claiming they are simply gifted don’t acknowledge the more difficult parts that can also come with it. HFA and ADHD don’t necessarily prevent intellectual careers. It massively depends how it affects that specific person and what their interests and strengths are. The hyper focus on certain topics can be a big plus in that situation. Diagnosis can be helpful. The support is needs based, but where he might not access sensory breaks or a one to one, he may have other needs that can be met in another way - for example, ear defenders, putty to fiddle with when listening in class etc.
I really doubt there is a huge amount of overdiagnosis going on. Private diagnosis can mean you never access NHS support for neurodiversity for your child. NHS assessment is a long road and hard to come by. For ADHD, if you’re being assessed for symptoms at home and not school, you need to do a workshop prior to the home assessment. Then after the home assessment, you can go on the waiting list (ours is 4y for ADHD). Prior to assessment, you have to do a parenting course. For ASD, you don’t have to do the workshop and parenting course, as far as I’m aware, but it’s a different pathway and a different waiting list again. You should still be offered courses to help you learn techniques.
School can put in support without diagnosis. They can do a one page profile to start with, if your son is struggling with anything. My DS masks at school, but the support there has made a big difference at home because he no longer regularly explodes. He also feels more validated because he knows people understand he may appear fine, but some things are harder.
What you want to avoid is getting to a crisis point before seeking help, because it’s just not possible to seek it fast enough and can be so awful for the child going through it. Teachers may be experienced, but they aren’t Paediatric psychiatrists. You see his behaviours at home and they see his behaviours at school. If he masks, then they may not see much that is suggestive of an issue. I wish I’d pushed for help earlier, but I allowed myself to be reassured by a teacher that my DS didn’t show signs (not their fault, they just weren’t able to see through the masking). Since then, I’ve learnt a lot more and the school has ended up being very supportive.
My DH and I are both ND. We also have family members who are. My DD is too young to tell for any HF or mild-moderate ADHD diagnosis (2y), she has little quirks that I’d mention in the paperwork should anything become more obvious later, but equally it’s fully possible she’s just 2yo and every toddler has quirks. However, by school age, if there’s any sign it’s possible, I’ll be getting her on the list asap because of waiting times.
Editing to add: Heritability of ADHD is 80%, so if you have it, there’s a strong chance your children might inherit it too. ASD can also have similar heritability, if not slightly higher.