Second, as alluded to earlier, your model is a kind of "package deal" comprising the imputation strategy as well as the missing data method. If your imputation method is so bad as to aversely affect its predictive ability, it is desirable to demonstrate this in terms of poor performing operating characteristics.
If the imputation method is poor (i.e., it predicts missing values in a biased manner), then it doesn't matter if only 5% or 10% of your data are missing - it will still yield biased results (though, perhaps tolerably so). The more missing data you have, the more you are relying on your imputation algorithm to be valid.
How much missing data is too much? Multiple Imputation (MICE) & R
By doing multiple imputation the proportion of ones in the long run will be the probability of being in that category. But you stick with 0/1 in combining analyses. Note that for PMM it doesn’t matter very much whether you use logistic regression or OLS for predicting the binary variable, as PMM just uses ranks of predicted values.
What imputation method should I use here and, more generally, how should I determine what imputation method to use for a given data set? I've referenced this answer but I'm not sure what to do from it.
I have seen Multiple Imputation by Chained Equations (MICE) used as a missing data handling method. Is anyone able to provide a simple explanation of how MICE works?
Typically imputation will relate to filling in attributes (predictors, features) rather than responses, while prediction is generally only about the response (Y). Even if imputation is being used to refer to filling in Y's the purpose is different; you're not using it for the primary purpose of getting a prediction for that Y.