Hernán and Robins in Chapter 8 of Causal Inference (Section 8.5) discuss an approach to correct for the bias introduced by censoring. It's an extension of your idea of evaluating the censoring data. In principle, it could get around the problem of distinguishing types of right censoring. It does rest on some assumptions, however.
The idea is to form a pseudo-population via weighting uncensored cases by their inverse probabilities (IP) of censoring. Then an uncensored case "accounts in the analysis not only for herself, but also for those like her" in terms of having the same treatment and covariate values. That requires an adequate model of the probability of censoring as a function of treatment status and covariate values.
The following assumptions are also required:
First, the average outcome in the uncensored individuals must equal the unobserved average outcome in the censored individuals with the same values of [treatment and covariates]...causal interpretation of the resulting adjustment for selection bias depends on this untestable exchangeability assumption.
Second, IP weighting requires that all conditional probabilities of being uncensored given the [covariate values] must be greater than zero....
The third condition is consistency, including well-defined interventions... the pseudo-population effect measure is equal to the effect measure had nobody been censored. This effect measure may be relatively well defined when censoring is the result of loss to follow up or non-response, but not when censoring is defined as the occurrence of a competing event.
Although the presentation in Chapter 8 is for outcomes without respect to time, Chapter 17 covers extensions to time-to-event analysis.
