OP I can totally get your exasperation, I have a "spirited" boy the same age. He's very full on and in some similar and some different ways to your child but I can understand how hard some days can be.
Def try and find help, keep trying till you find the right thing. If you have a good GP I'd start there and get a referral to a paed and maybe a child psych.
My almost 3 year old is extremely hyperactive and while he can concentrate on things he wants to, he's not easy. He's a very skilled climber, tries to escape constantly from the car seat, is a bolter, and general just extremely high energy. Im yet to meet another child to rival him. He can get up at 6am and literally has to be wrestled to sleep by 10pm. He's on the go constantly, I've had 2 GP's tell me he's the most full on kid they've met and that was at 18 months. As a baby every single person would comment on how "wired" his eyes looked.
Diet i think can play a big role though sounds like you're already onto that. We are the same can't really give anything that comes in a packet (though I still let him have occasional treats, just have to be prepared for tornado mode for awhile). Has to be all clean/homemade food. We're about to see a new paed to talk about a more strict exclusion diet maybe limiting wheat and dairy.
A child psych might be a good place to start? If your GP can recommend a good one? My son is also a late talker and i think that can make their frustration greater.
Don't assume a dr is just going to put your kid on meds, there are lots of things im sure they can suggest to try.
Do you find he has even one activity that he really prefers?