We revisit the family of goodness-of-fit tests for exponentiality based on the mean residual life time proposed by Baringhaus & Henze (2008). We motivate the test statistic by a characterisation of Shanbhag (1970) and provide an alternative representation, which leads to simple and short proofs for the known theory and an easy to access covariance structure of the limiting Gaussian process under the null hypothesis. Explicit formulas for the eigenvalues and eigenfunctions of the operator associated with the limit covariance are derived using results on weighted Brownian bridges. In addition we provide further asymptotic theory under fixed alternatives and derive approximate Bahadur efficiencies, which provide an insight into the choice of the tuning parameter with regard to the power performance of the tests.