Machine Learning with intended missing values
The common case of missing values for which data is imputed or removed assumes that missing values appear randomly in the data, so the absence of value has no relevance to the task.
From your description, in your data the fact that a value is missing is significant by itself. So I'd say that yes, it makes sense in this case to represent this information as a categorical variable. Note that it can be represented as a special value for the score feature indeed, but it doesn't have to be the same variable.