Actually, what you have described is the definition of "Current Status Data". The idea being that each subject is only observed once, and all we know is that an event has occurred or has not. This results in all of our data being either left censored (i.e. at time of observation, we see that the event has occurred) or right censored (at time of observation, we see that the even has not occurred).
As such, you should use methods for interval censored data to analyze your data. Standard interval censored data allows you to define an interval for the time of event for each subject such that
$t_i \in (l_i, r_i)$
where $t_i$ is the (unobserved) event time, and $l_i$ and $r_i$ are the lower and upper limits of an interval that you know to contain $t_i$. So with current status data, if we inspect subject $i$ at time $c_i$, and the event has occurred, we record their interval as $(0, c_i)$ (i.e. we know the event occurred before $c_i$), but if it has not occurred, we say $(c_i, \infty)$ (i.e. all we know is that the event has not occurred by time $c_i$).
Extremely biased here, but I would recommend the R-package
icenReg, which allows Cox-PH and proportional odds regression models for interval censored data. Another great package is
interval, which allows for log-rank tests (among other functions), but not regression models. Finally, the
survreg function in the standard
survival package allows for fitting a fully-parametric accelerated failure time model.
The bias towards
icenReg is that I am the author.
Interestingly, your dataset sounds almost identical to one of the earliest publications involving current status data. See
Hoel D. and Walburg, H.,(1972), Statistical analysis of survival experiments, The Annals of Statistics, 18, 1259-1294
In fact, this dataset can be found as an example in