‘Well-rounded’ and ‘dimensional’ are terms I wish would die. They are not helpful terms. All they do is confuse the hell out of writers. ‘Fleshed out’ is probably a better approximation of what people actually mean but it still doesn’t really nail it.
IMO there is only one thing a character needs in order to function: understandable, believable motivation. Everything else flows from that. We have to believe - or at least to know and care - why they do things.
Take Hannibal Lecter on the TV show HANNIBAL. On the face of it, nothing he does makes sense. e.g. He’s kind one moment, vicious the next. Calm then violent. Charming then steely cold. A passionate lover and a cold-blooded killer. He is ‘multi-dimensional’ almost to the point of nonsense. But it works because we know his motivation. The writers actually have him say it aloud a number of times, so it’s beyond doubt: “I wanted to see what would happen.” Hannibal does Hannibal because he believes he’s like God. Once we understand that, everything he does makes sense.
Sarah Connor in TERMINATOR 2: JUDGMENT DAY is a great heroine not just because she kicks butt, but because she believes that, given her circumstances, this is the only Sarah Connor it is possible for her to be. She is not kicking butt because it’s cool or to make her feel good about herself. Sarah doing Sarah is a matter of human survival.
So Sarah Connor in T2 isn’t actually all that ‘dimensional’. She has one dimension: get in my way and I will end you. But I imagine most would agree she’s a wonderful character and a ‘strong woman’. In fact, it’s her initial single-mindedness that enables her to have such a great character arc.
So yeah, I think motivation is more crucial than how many ‘dimensions’ a character has.