The coalescent is a foundational model of latent genealogical trees under neutral evolution, but suffers from intractable sampling probabilities. Methods for approximating these sampling probabilities either introduce bias or fail to scale to large sample sizes. We identify a class of functionals of the coalescent which describe the variance of estimators from classical importance sampling algorithms, and which have tractable infinite-sample limits. These functionals provide the first mathematical descriptions of the performance of some seminal coalescent inference methods, and reveal that coalescent importance sampling differs markedly from the behaviour of (sequential) importance samplers in more standard settings, with or without resampling.