Multiple imputation can be fully Bayesian. Chained equations (CE in MICE) doesn’t form proper joint distribution, so that can be considered not to be principled. Not all missing value cases are simple and we still often don’t have enough computation power. MICE is flexible for many missing value problems and scales well and the single models can be Bayesian models. Joint latent variable models would be alternative which would define proper joint distribution. The inference is a bit challenging, but they could be more popular. As Stan doesn’t allow discrete parameters, imputation with (Bayesian) MICE is very useful.
This is much easier case than missing values in X.
I recommend this, too, unless Y is multivariate and only some values for each row are missing. It can still be possible that the likelihood factorizes and you can just drop likelihood terms for the missing cases.
Yes, chained equations of MICE are not needed if only on variable has missing values.