We propose a novel approach to uniformity testing on the $d$-dimensional unit hypersphere $\mathcal{S}^{d-1}$ based on maximal projections. This approach gives a unifying view on the classical uniformity tests of Rayleigh and Bingham, and it links to measures of multivariate skewness and kurtosis. We derive the limiting distribution under the null hypothesis using limit theorems for Banach space valued stochastic processes and we present strategies to simulate the limiting processes by applying results on the theory of spherical harmonics. We examine the behavior under contiguous and fixed alternatives and show the consistency of the testing procedure for some classes of alternatives. For the first time in uniformity testing on the sphere, we derive local Bahadur efficiency statements. We evaluate the theoretical findings and empirical powers of the procedures in a broad competitive Monte Carlo simulation study and, finally, apply the new tests to a data set on midpoints of large craters on the moon.