(1) Should note first that the weights we are talking here are probabilitiy weights, not the weights that are provided by the survey conductors.
(2) the result of Assumptions 5.1 and 5.2 in Wooldridge (2002) coincides with that from the missing-data workshop in London. That is, the unweight M-estimator (complete case analysis) is valid ( unbiased and consistent) when R depends on X or is independent of both Y and X. (X here must be observable (not a latent variable like ability) but is allowed to be missing when R=0)
Missing-data workshop ( the introductory one) maintains that if R depends on X, the adjusted mean of Y for X is unbiased. ( adjusted mean is the coefficient of that X in a regression model) Also, if R depends on another random variable, say, S where S is independent of Y and X, then the result is also unbiased. Note that, in their example, S is uniformly distributed in (0,1). Can S be normally distributed???
Note that, from this course, the complete-case analysis is biased if R depends on Y or on both Y and X.
Wooldridge (2002) adds another condition to the conditions mentioned above. He shows that correct specification of the distribution of Y|X or that of mean function is also required.