As the answer from @AlexJ says (+1), you do have data that can be reformatted in a way that can take advantage of well-documented statistical survival functions. What can be surprising is that, when there's at most one event possible per individual, the individual identities often don't matter.
Recall the formula for the Kaplan-Meier estimator. It's calculated at each event time $t_i$ based on the number that had the event at that time, $d_i$, and the number that were at risk of the event, $n_i$. The estimated survival at time $t$,$\widehat S(t)$, is then:
$$\widehat S(t) = \prod\limits_{i:\ t_i\le t} \left(1 - \frac{d_i}{n_i}\right).$$
There's nothing there that requires knowing individual identities, just how many were at risk and how many experienced the event at each time. It's only in more complicated cases (e.g., more than one event possible per individual, some parametric survival models with time-varying covariates) that you need to keep track of individual identities.
Replicates
In your experimental design, however, you probably do need to keep track of the "Replicate" associated with each individual. That will allow you to take potential correlations of outcomes within a Replicate into account. I suggest that you re-name the Replicate values, now all numbers 1 to 6, to include both the Treatment and the Replicate number, so there's no confusion that Replicate 4 in Control is the same as Replicate 4 in Treatment 5: e.g., use names like "R4C" and "R4T5" to keep them clearly distinguished.
"Interval-censored" data
The format that @AlexJ recommends represents the data as "interval censored"; that is, you know that the event occurred sometime between the two limits of the time interval, but not exactly when. In that format, the 0/1 status data you are thinking of is effectively replaced by the time value at the end of each interval: if it's finite, there was an event during the interval (status = 1), if it's infinite (Inf in R) the time to the event is right-censored (status = 0).
Need for regression model
The problem with a Kaplan-Meier analysis in this case is that it will be hard to distinguish treatment differences very well when you have so many treatments. The usual "log-rank" test done as part of such analysis can tell you if there are any differences among treatments, but not which ones are significantly different from control or which differ from each other. To get those types of results it's best to build a regression model of some type.
A Cox proportional hazards model is a frequent choice. In your situation that would not require keeping track of individuals, either. There are tools available in the R icenReg package for Cox models on interval-censored data. I'm not sure, however, that those tools can take the "Replicate" values into account in an efficient way.
One choice, often made in practice even though it's not theoretically exact, would be to perform a standard Cox model where the event time is recorded as the end of the time interval even though you don't know exactly when the event occurred. You can think of that as a model of when you observed the death rather than when it occurred. A cluster term in the Cox model could then take "Replicate" values into account.
With your current data setup, you can generate a new data table for that pretty simply. For each time point, for each row in your current data, create a number of rows in the new table equal to the difference in survival since the previous time point,* copying over all the information about Treatment/Replicate, setting the Time for those rows to be that time point, and the Event marker to be 1. At the last time point, make copies with the Event marker of 0 instead for the number that survived at last observation. That way, you end up with a number of rows equal to the original number of animals, with the last observation time and the status at that time (even though this doesn't distinguish which animals are which).
Discrete-time survival
As discussed on this page, however, with so few time points you might be better off with a discrete-time survival model. That's just a binomial generalized linear model with data in a long "person-period" format: one row for each individual at risk during each time interval, with indicators of the time interval, the treatment, the replicate, and 1/0 for event/no-event during the time interval.
If you have a new data table as recommended above, with one row per individual including the time point and status at the last observation, then you can use the dataLong() function in the R discSurv package to generate the (very long) person-period data format. A choice of timeAsFactor=TRUE will be most similar to the Kaplan-Meier and Cox models, in that there is no assumption about a numeric form of baseline survival over time. That generates the long-form data with a "timeInt" value representing the time interval and a new outcome variable "y" for the status of the individual at the end of that interval.
With such person-period data, you could use a binomial generalized mixed model to take the Replicate values into account, treating Replicate as a random intercept. That's implemented by the glmer() function in the R lme4 package. To be closest to a Cox model you could specify the "cloglog" link instead of the default "logit" link used for binomial logistic regression. The model could look something like this:
discreteModel <- glmer(y ~ timeInt + Treatment +
(1|renamedReplicate),
data = dataInLongFormat,
family = binomial(link = "cloglog"))
Then you can use post-modeling tools like those in the R emmeans package to evaluate specific comparisons of interest. For example, you can evaluate the difference of each treatment from control if you specify "trt.vs.ctrl" contrasts in that package.
*Before you do this, double-check your data so that you don't have any "resurrections" like you show for Replicate 6 of Control in your sample data.
