In Mathias' paper Hilbert, Bourbaki and the scorning of logic (see https://www.dpmms.cam.ac.uk/~ardm/hbslmag2.pdf), he mentions Hilbert's $\varepsilon$-calculus (and thus Bourbaki's $\tau$-calculus) is not suitable for incomplete systems. But he did not elaborate if there is any real problem (like inconsistency or incapability) with it.
Obviously Hilbert hoped to use $\varepsilon$-calculus to prove the completeness of systems like ZFC, and failed as we know. But this does not means $\varepsilon$-calculus itself has any problems.
So, is Mathias' criticism out of his personal flavor of logic and educational reasons, or there is any real problem for $\varepsilon$-calculus in incomplete systems?