I would agree that a menstrual cup definitely helps against leaks. There are loads of brands these days, so if one doesn't suit, another might. I found it took me till my second period with it to be really confident with using it. I do find that while leaks are less likely, on my heaviest days - it can still fill up, but you do get longer than with a super-plus tampon, so it's easier to manage, but there is still some risk of leaks if you go too long.
It's still a good idea to see your GP to get your iron levels checked, and to see if there's anything like endometriosis or fibroids or whatever going on. It can be helpful if you've hot some idea of flow, which it sounds like you have - how quickly you get through protection, any clots (and size), pain levels and anything else.
My GP said, "no woman in this day and age should have her life disrupted by periods, and there's a scale of treatments we can try until we're managing it." But other GPs are less helpful and will try to fob you off with all women go through this, it's normal, just live with it. If you have one like that, get another GP.
And I wouldn't worry about the 3yo - I wouldn't go put of my way to let her see it, but if she does, just give a basic explanation about how you're fine and it's a normal thing.