With a binary target and categorical features, logistic regression can be viewed as a type of log-linear model or contingency table analysis. In chapter 10 of Wickens book, Multiway Contingency Tables Analysis for the Social Sciences, approaches to adjusting degrees of freedom and model results for the presence of structural zeros are discussed. Wickens main point is that "data tables with structural voids lack a complete factorial structure," a necessary requirement for logistic regression tests of independence, where "impossible cells assert themselves as dependencies," pps. 246-247.
One option is to step back from considering time as continuous. Discretizing time would place the model design into the form of a contingency table. This permits one to ignore the missing cells and test for quasi-independence of the valid cells only.
Wickens suggested solution is to employ an iterative proportional fitting algorithm to construct the maximum likelihood estimates which he goes through on a step-by-step basis -- much too involved for a response on this blog. In essence, in constructing the test statistic, the structural zero cells are ignored and degrees of freedom are adjusted or reduced for the missing cells, devising a new test statistic based only on valid cells.
Apologies if this is too involved for a stats newbie but, in point of fact, your problem is not a big one and is quite readily answered. The only issue is that these methods go well beyond the full factorial chi-square tests of independence dealt with in stats 101 courses. There are many, many references and resources on log-linear models, contingency table analysis, and categorical data analysis. While Agresti's books are among the standard, stats "go-to" resources, I prefer Wickens for the greater clarity and lucidity of his writing. To be able to fully address, understand and answer your question, you need to drill into those references.