In my data I have $n_1$ people who had an "event" and $n_2$ people who did not. Cases (those with an event) were oversampled substantially (the true prevalence is probably more like 1 in 10000). If it matters, $n_1 = 70$ and $n_2 = 250$. For all of those who had an event, I also know how long it took from baseline for the event to occur.
I also have a quantitative predictor measured on each person.
I want to see how that the quantitative predictor is related to the "survival time". I know if case-control status were based on something other than the variable that generates the survival time, it could just be treated as a binary predictor. But, that's not the case here. Given the wild oversampling of cases (and thus a biased sample of survival times), I feel there's something not quite right about doing survival analysis as though this were a random sample of survival time/covariate pairs.
Ignoring times and only focusing on the binary indicator of whether or not you had an event, I do know that logistic regression is still OK in this case for estimating Odds Ratios. Is that the best I can do in this case? If so, that is OK, I just don't want to look stupid for not doing survival analysis if it's feasible to do so.