I must say, playing with a toddler that has the Independent trait quite annoying, but there are a few side benefits to it.
Now that I've played a couple of toddler's with this trait, I rather like it. I did have to learn to not micromanage those toddlers, which was an adjustment. They do a fine job of taking care of themselves and bug adults easily for what they need. Free range toddlers. When there are four toddlers, and the mayhem of different traits like you say, that one is the most safely ignored. Just leave some low spoilage food out, and it will probably not only be alive but have met all its needs after you put out the Clingy and Fussy fires, while the Wild child kept interrupting for dance help 12 times.
The best incident yet: Nightmares. Most toddlers wake at least one adult for comfort, some wake several! The independent one decided to go grab a cookie from downstairs, since he was also hungry, pottied himself and by then was no longer scared and put himself right back to bed. Fantastic midnight adventure and no adults the wiser, except for the missing cookie.
Yes, they do get the angry moodlet from some interactions but it makes me laugh. The mouseover text for being lectured is so hilariously ... toddler. It ends with "Who do they think they are!".