If it is purely a binary classification problem (class A vs. class B), then the benefit of the F-score is primarily for characterizing performance over an unbalanced data set (more instances of one class than the other) and your question/concern is more relevant. The Wikipedia page for F-score states
"Note, however, that the F-measures do not take the true negatives into account, and that measures such as the Phi coefficient, Matthews correlation coefficient, Informedness or Cohen's kappa may be preferable to assess the performance of a binary classifier."
But if the classifier is intended to be a detector, one is usually more interested in performance with respect to the target class (Positive) than the non-target class (Negative). Furthermore, the target is often the one that is under-represented in the data set. In that context, I think it is more intuitive to want to know what fraction of targets are detected (recall) and how reliable/confident each detection is (precision). While knowing how good the detector is at not detecting non-targets (negative predictive value) can have value, it is not a very insightful quantity to deal with when trying to characterize the performance of a target detector with an imbalanced data set.
In short, the F-score tuning parameter ($\beta$) provides a more intuitive way to balance the importance of detecting all the targets (high recall) with the importance of having detections with high confidence (high precision). Note also that the F-score can be written in terms of Type I and Type II errors (see the Wikipedia link above).