I like the concept of a proper scoring rule (note that quadratic loss is one). However how often is one interested in extracting a subjective probability? In this case the object of interest seems to be the outcome rather than the probability.

Overall I suspect that the loss function doesn't matter much - you either find the "structure" or not. However, it is a curiosity that in statistics, setting the model for structure (probability) leaves no freedom in modeling the loss (which is minus the loglikelihood). What happens if I want to fit a Poisson model, but I'm interested in minimizing predictive least square error? Is that impossible? Trivial? Meaningless? I'm not sure.