should we include or exclude a variable in a logistic regression based on the description below?
You may want to add an indicator variable, 0 if the event has happened and 1 if it has not.  In a linear model, the coefficient on the original can fit to the "real" slope, while the coefficient on the indicator "fixes" for the missing values (and whatever imputation you use).

https://stats.stackexchange.com/a/105258/232706
https://datascience.stackexchange.com/a/64709/55122
https://datascience.stackexchange.com/a/69841/55122