If you are certain that this is not normal behavior, go with your gut. Doctors (MD's) are utterly useless for this sort of thing. Five minutes of observation simply doesn't give them enough data points to be able to tell you much of anything.
Regarding parenting classes and normal disciplinary approaches: try a new strategy with absolute consistency for a solid week. If you haven't seen any significant changes in a week, it is not an appropriate strategy for your child. Try something else. No it probably won't work better if you just try harder and longer, no matter what the "experts" say. If you find something weird that works, go with it, no matter what the "experts" say.
But what you're describing sounds like a kid under some type of extreme stress. Your child sounds like he's deeply, constantly frustrated by something he cannot fix and cannot communicate. Because he is three, he expects you make it better. Since you are the all-powerful adults, he subconsciously assumes you are intentionally allowing him to remain uncomfortable. That's where the anger is coming from.
It's not likely to be resolved by normal discipline. It would be like trying to discipline a child to stop screaming....when there's a nail in his foot! Your job now is to identify and resolve that stress. This is pretty much a series of trial and error. Start with the basics and work outwards.
Your core skill is observation. You're looking for patterns. Since you spend the most time with him, you are the most qualified to do this. Start a journal. Note time of day, good times, bad times, and any environmental factors you think might be relevant (food, location, people, etc.) Create the shortest shorthand you can think of, since you don't have a lot of free time. Ask others for their observations as well.
Professionals are unlikely to be answer the question "what's wrong with my child?" Professionals are very useful for answering questions like "Are my child's sleep patterns within the healthy range for his age?" or "Does it appear he's dealing with some kind of sensory processing disorder?" Once you can get a specific question, the right professional can be gold in helping you come up with an accurate answer. But you have to develop the specific question yourself.
Things to think about:
Sleep: He should be getting about 12 hours of quality sleep. If not, there's a problem. Does he have a regular sleep schedule? Does he appear rested in the morning? Does he have dark circles under his eyes? All of the behaviors your describe can be caused by severe sleep deprivation.
Breathing: Is he breathing well. Are you sure? Two doctors told us my son's breathing was fine. Two doctors were wrong.
Sensory problems: read The Out-Of-Sync Child. See if anything sounds familiar.
Allergies and environmental illness: Food allergies, mold, dust mites, pollen, etc can put pretty severe stress on a child (or an adult).
Pain: If your child is in chronic pain they will not usually complain of pain. It becomes normal to them and they can no more describe it than a fish can describe water. Doctors are often very quick to assume a child's behavior problems are due to poor parenting techniques. Sometimes this makes them less diligent about checking health concerns. If you think there is a problem in this area, be persistent. Be sure to check teeth too.
Nutritional deficiencies: Certain people have abnormally high needs for certain nutrients. Even if your child has a fairly healthy diet, it might be worth experimenting with increased doses of various types. Work closely with a nutritionist if you try this approach, to avoid toxic overdoses.
And......whatever else you can think of. It may be a simple fix, once you find the problem. Or, there may be several different issues stacked up on top of each other. Whichever it is, the most difficult part is usually identifying the problem. Once you know what you're dealing with, you can fix it or make appropriate accommodations.
My son is my "onion boy". We were dealing with similar issues with him at age three. Five years later, we're still peeling back the layers of what he's struggling with. So far we've found breathing problems (resolved), severe sleep deprivation (resolved), black mold in the house (resolved), speech problems (improving), sensory problems (improving), visual problems (starting therapy now), and dyspraxia (starting therapy now). We have not yet identified all the layers, and will be doing another round of assessments this year to find the next layer.
He is truly a joy to live with now, but he still gets frustrated more easily than many kids. I explain to people that for him, most tasks and interactions take a level of concentration and effort akin to a lawyer studying for the bar exam. When he withdraws or gets uncooperative, it's because he's exhausted. You would be too!