Anecdotally, I've often heard it said that unbalanced designs are problematic for ANOVA but not for mixed-effects models. For example, in this question someone had unbalanced data (very few observations at the predictor level 'Day 8'), and was planning to do a repeated-measures ANOVA. Then someone else gave this answer:
"I would recommend looking into mixed-effects models (also called random-effects models, multilevel models). They don't require balanced data--i.e., they handle missing data on the dependent variable--so you wouldn't have to drop your Day 8."
The first and third answer to this question also make similar statements, without providing an explanation.
Could someone please explain if the reasoning behind such arguments - why are unbalanced data not a problem for mixed-effects models, but they are for ANOVA? And how about ordinary regression?