Survival bias occurs in retrospective studies where inclusion is in some sense outcome dependent (through outcomes or their moderators) but is treated as representative of a population at-risk at baseline. Your description does not give any details suggesting survival bias is an issue here.
Censoring leads to a different type of bias, censoring bias, when not properly accounted for. Your analytic plan of using a Cox model does properly account for censoring thus eliminating censoring bias. Despite that, censoring does reduce the power of an analysis. Suppose you monitor 5,000 people but only 10 experience an outcome (death or other), the Cox model does not afford much more power than a survival analysis of only 10 people.
Your description of the exposure is not exactly clear to me. It sounds like participants are eligible to participate in the study only if they have not yet begun a certain therapy. After a period of self-determined time, they begin a therapy. You then follow participants for an outcome (at the time of which they may be either on or off such a therapy).
This is an analysis that should be done using time-varying covariates with some caveats. When I enter the study, irrespective of calendar time or age, my survival "clock" is at time 0. If I initiate therapy at day 10, and then die at day 20 I contribute two correlated observations to the sample: the first I live 0-10 days with no therapy and am censored at time 10, the second I live 0-10 days and die at time 10. The clock resets when I initiate therapy. Frailties are the Cox model equivalent of random effects that allow you to account for clustered observations in such a format. If age and/or calendar year are significant predictors of survival in such a study, you should consider adding them as covariates in the model.
The caveat to time-varying covariates is as follows: initiation of therapy almost always depends on latent disease state. Patients whose initial hospitalization requires high acuity will initiate therapy more quickly, and likely die more quickly even if the therapy is beneficial. This leads to use-bias. If you measure indicators of disease state longitudinally (like blood pressure, physical functioning, or other), latent variable models or marginal structural models may be used to reduce such a bias.