I'm just starting to learn about missing data imputation methods, and I'm confused. In every introduction I've read, the author starts by describing listwise deletion and says that it's a bad idea because it reduces your N. They then explain methods you can use to fill in missing values so you can do your analysis with all your participants. That makes it sound like the purpose of imputation is to avoid throwing out other data points that you did observe.
Question 1: Is imputation useful only because it lets you use the values you observed? Or do the imputed values themselves improve the analysis?
For example, let's say I'm analyzing a repeated-measures dataset with a linear mixed model, and some participants are missing some timepoints. (Let's also say that the data are missing at random). Linear mixed models already work with incomplete data. Is doing imputation on the missing data points still appropriate, even though their missingness isn't causing me to throw out any other data?
Question 2: What happens if I use full information maximum likelihood to impute values for the missing data points? It seems like that does provide some additional information for my model -- I get to include the portion of the missing data points that can be inferred from other variables in my dataset, even if those variables aren't part of the model I'm testing. But is that actually legitimate?