This sounds like my boy. Exactly like my boy. He was recently diagnosed.
But he got suspended from before and after school care from the time he was 2. He would draw on walls, kick things, push, hit, roll around, run around, hop, skip, jump, blurt out, and everything inbetween. The diagnoses in my case made so much sense, but I wasn’t trying to allow someone to tell me that he had ADHD at 2 because sometimes it’s just too early to tell.
I did say though, if the issues persisted into Kindergarten I’d get him tested. Now we’re working on regulating medication, behavioral therapy, a 504 plan, etc. His teacher and administration have helped immensely with everything as well. His behavior in school has changed, and he’s no longer labeled the “problem kid”. With the diagnoses came understanding and a desire to help.
I say get him evaluated. The worst case is that he has ADHD, and that’s not that bad.