Missing Values: Indicator, Impute, or Let the Model Handle It

A lending dataset has annual_income missing for 18% of applicants. Investigation shows income is missing far more often for gig-economy applicants (who often don't have a single employer to report to) than for salaried applicants.

1. Is this data missing completely at random, at random, or not at random? Why does the distinction change what you should do?
2. Design a missing-value strategy for (a) a logistic regression and (b) a LightGBM model.
3. A colleague proposes dropping all rows with missing annual_income because "we want complete, high-quality data." What's wrong with that?