Major advances have been made regarding the utilization of artificial intelligence in health care. In particular, deep learning approaches have been successfully applied for automated and assisted disease diagnosis and prognosis based on complex and high-dimensional data. However, despite all justified enthusiasm, overoptimistic assessments of predictive performance are still common. Automated medical testing devices based on machine-learned prediction models should thus undergo a throughout evaluation before being implemented into clinical practice. In this work, we propose a multiple testing framework for (comparative) phase III diagnostic accuracy studies with sensitivity and specificity as co-primary endpoints. Our approach challenges the frequent recommendation to strictly separate model selection and evaluation, i.e. to only assess a single diagnostic model in the evaluation study. We show that our parametric simultaneous test procedure asymptotically allows strong control of the family-wise error rate. Moreover, we demonstrate in extensive simulation studies that our multiple testing strategy on average leads to a better final diagnostic model and increased statistical power. To plan such studies, we propose a Bayesian approach to determine the optimal number of models to evaluate. For this purpose, our algorithm optimizes the expected final model performance given previous (hold-out) data from the model development phase. We conclude that an assessment of multiple promising diagnostic models in the same evaluation study has several advantages when suitable adjustments for multiple comparisons are implemented.