Is "sensitivity at fixed specificity" a valid metric for comparing different classifiers? For a given dataset, a common way to compare 2 classifiers is to compare their average validation accuracies using cross-validation.
Is it valid to replace the accuracy with other classification metrics that I care more about? For example, say I care about the sensitivity (recall) at a given specificity level (say 0.99). Is it still valid to compare A and B by computing the average sensitivities using cross-validation? (for each fold, train the model, plot the ROC curve, get the recall for specificity=0.99)

Yes, you can choose any metric of interest while you're validating your model.  This can be a specific metric of your choice, like sensitivity value at a specific specificity as you proposed, if you think that best suits your need; or area under ROC, precision/recall curve, F1-score etc. The space of possible choices is not limited to accuracy.
