Can I deal with a missing not at random column by creating a new column? (Feature engineering)
Creating a new variable as you described would be redundant as it is a function of the other two variables. In other words it is not adding any information.
The below suggestion assumes the model cannot deal with missing values, but a lot of the best models (ex. xgboost is typically one of the best for classification) will deal with this in a smart way for you.
Just fill the missing values via imputation, ex. by using the mean score or training a model using all the other features to predict the scores, and then predicting the missing ones. The model will be able to take into account that some users didn't complete the action, so it won't just treat them as if they got the imputed score, since their value in column A is 0 and that is useful information.