All schools are equipped and qualified for sen. That's why they have a senco and a budget for it. But you have identified the issue yourself. You have to be the driving force (this is what your friend was trying to say, I think) because the pressure on the system for that funding is fierce. And the children competing for it have varied and in some cases severe needs. So the onus is on you to prove that ds needs access to it.
Right. So you think it's ADHD or dyspraxia. You make a long list of how he fits each, and you go back to the paed or gp. If snap testing has already ruled out ADHD, and the paed won't follow up, then either you have to pay for a private test through a child psych, or you have to leave it for a year, collect more evidence, and go back and ask for a repeat (ad infinitum). Probably three years if school persist?
Dyspraxia - as you haven't sought help for that yet, go to the paed with a list of issues which require assessment. Ask them to refer for an OT assessment. Waiting list are usually quite long - they can be in excess of six months just for assessment, and then some areas have a wait list for therapy, if ds fits into whatever bracket they deem to treat in your area. Or you could pay for a private OT assessment. Sometimes children with dyspraxia aren't given much in the way of therapy, sometimes they are, depends on severity. Some areas have great OT and physio programmes. But again, only for the most severe.
If he has already been seen by the paed and discharged, there is v little chance of the gp doing the referrals. The one usually cancels the other out in the grand scheme of things. Usually the the developmental paed that makes the call rather than the generalist. But you might find a helpful gp.
And once you've got the reports that say ds has x dx (or diagnoses) then you have to liaise to school to work out what support he needs and they are willing to provide.
At a guess, I'd say you spent a grand and got a wobble cushion and a prescription for meds. And an acknowledgement that he's dyspraxic not clumsy. Anything else will still be up to you to fight for.
I'm being deliberately blatant about how hard this is going to be. A label (or two) will make very little difference in real terms. It will just give you a piece of paper to go into battle with. As your friend knows, your interaction with school is what will make the difference for ds. A dx will help, but it won't solve the problems.