There is absolutely no advantage to using e.g. logistic regression in the case you describe, if you actually have the individual records with survival and censoring times. The type of situation you describe happens all the time in practice (even lower percentage of records with an observed are common) and this is not necessarily bad experimental design, but sometimes simply unavoidable (e.g. randomized cardiovascular outcome trials).
And there are a lot of downsides. If there are losses to follow-up prior to the end of the planned observation window (or it was a window of different length for different individuals as in event-driven clinical trials), then survival analysis is the appropriate way to account for these differences between records. Doing so is particularly critical, if some variables of interest in your model are also associated with different censoring patterns. When you use binomial data methods, you do not really have a way (or requires ad-hoc fixes that are at best approximately valid) to predict what would happen during observation periods of a different length.
You can actually think of logistic regression and other binomial data methods as a survival analysis that throws away all information about exactly when events happened and that also assumes that there is no censoring prior to the end of the planned observation window. When viewed this way, it is sort of clear that the first part primarily entails some loss of information (not so bad, if you have an exponential event time distribution with a really low rate, extremely bad if all records have an event), while the second assumption brings in a substantial risk of bias when it is not true (alternatively it is also enough to assume that all records need to have the exact same distribution of censoring times).