Using multiple imputation on components of a derived variable

Yes, a suitably applied multiple imputation can make a lot of sense in such a setting. In fact, it can often be a lot easier to impute more fine-grained scores as opposed imputing event times directly.

Similarly, with questionnaires where some questions may be unanswered at each timepoint, it is often a good idea to impute for each question first and then to calculate the overall score, while setting the overall score missing if not all questions are answered and imputing the overall score is a lot less efficient.

In the end, yes, you will get for each imputed dataset with imputed scores a corresponding dataset with imputed event indicators/times. Your intuition is right: you then analyze each of these imputed time-to-event datasets and combine the results using Rubin's rule. The main question here is: on what scale to combine (and whether we can get valid/useful standard errors on this scale). You want a scale where the normal assumptions in Rubin's rule are about right. So, e.g. the log-hazard ratio scale, if you are interested in hazard ratios (and similarly some transformed scale for survival curves).