For example, if I token is deleted does it have a duration of zero? It's not 100% clear to me that this should be the case. If we assume it is, and you fit some sort of zero-inflated model of duration, that is the same as a logistic regression for deletion + a seperate continuous model.

This is because the probability of a single value in the continuous model is zero, so you don't need to model the zeros (deletions) as coming from two processes like you would with a discrete model like a zero-inflated gamma-poisson.