We've been that family, not just at Disney but at other places too. It happens! Mine are now 4, 10 and 12 but when my 10 year old was 3 (almost 4) we had some epic Disney moments. She has always been the "challenging" one.
She was very overtired one evening and we were scheduled to do the Luau at the Poly. They bring you to this holding area of sorts before they walk you down to the stage/seating area. She was being so naughty, not listening, telling me "no" repeatedly and then started hitting me in the arm over and over. I remember there was this one woman in particular who kept looking at us. In that situation, there was really nothing I could have done, we were in extremely close quarters with other guests and if I reacted to my 3 year old the way I wanted to, it would have been a complete scene which I was trying to avoid. But this woman kept giving me the evil eye as if she couldn't believe I was letting my daughter act this way. I'll never forget it. Thinking back I wish I had said something to the woman but at the time, I just wanted the moment to pass. Anyways, we finally got to the table and after she filled her belly with food, she passed out in my lap and slept through the whole thing.
Long story short, kids are kids and especially at Disney where they are out of their schedule, there is so much stimulation, they get hot, tired and overwhelmed easily, there are bound to be tantrums. It's all about how you react to them and trying to anticipate/see those moments coming when you can. We have learned over the years to follow the kid's lead. We make sure they are well fed and get the rest they need while we are there. We don't push them too far and we are a mid-day break family. You're never going to completely avoid bad kid behavior no matter how hard you try (sometimes it's for no reason at all), but I never do commando with my kids and if they say they don't want to do something at Disney, we don't force them. We go often so we know that the rides/shows/food and entertainment or whatever it may be will be there next time.
We've all been there!