Max-stable processes are increasingly widely used for modelling complex extreme events, but existing fitting methods are computationally demanding, limiting applications to a few dozen variables. r-Pareto processes are mathematically simpler and have the potential advantage of incorporating all relevant extreme events, by generalizing the notion of a univariate exceedance. In this paper we investigate the use of proper scoring rules for high-dimensional peaks-overthreshold inference, focusing on extreme-value processes associated with log-Gaussian random functions, and compare gradient score estimators with the spectral and censored likelihood estimators for regularly varying distributions with normalized marginals, using data with several hundred locations. When simulating from the true model, the spectral estimator performs best, closely followed by the gradient score estimator, but censored likelihood estimation performs better with simulations from the domain of attraction, though it is outperformed by the gradient score in cases of weak extremal dependence. We illustrate the potential and flexibility of our ideas by modelling extreme rainfall on a grid with 3600 locations, based on exceedances for locally intense and for spatially accumulated rainfall, and discuss diagnostics of model fit. The differences between the two fitted models highlight how the definition of rare events affects the estimated dependence structure.