Why use a stratified log-rank test if we balance over strata anyway? When comparing survival data over time, we can use the log-rank test. An extension of this is the stratified log-rank test, which adjusts for variables.
Let's say we wanted to adjust for the effect of sex, when comparing two treatment groups, we could then presumably use this stratified test. However, would this make sense to do if we balanced the sex over both treatment groups at the randomization stage? Surely you would then be adjusting for sex twice? (once when assigning the treatment groups, and again at the analysis stage). Does this make sense to do? Or should you do one or the other?
Intuitively this makes no sense to me! Please could someone explain?
 A: There are several reasons why it is often a good idea to control for a covariate (like sex) even if you balanced the treatment groups to start with.
One is that you might not have balanced completely, so this gives extra control.*
A second reason, even with complete balance, is when the covariate itself might be associated with outcome. If you ignore it you will have large within-group error outcome estimates, as much of the variance in outcomes within each group might be due to the covariate (e.g., men tend to die at younger ages than women). With higher within-group error estimates, you have less power to detect between-group differences. You might find the FDA draft guidance on covariate adjustment to be enlightening.
A third is that there might be a difference in treatment effects depending on the covariate value. Unless you incorporate the covariate into your analysis properly, you can't evaluate that possibility.
A web search for "covariate adjustment randomized trials" can provide many further details.

*There's an analogy in non-randomized studies, in which group covariate balance is attempted via weighting cases by their inverse probabilities of receiving the treatment, based on their covariate values. Then further adjustment for the covariate gives you a "doubly robust" estimate, in that only 1 of the balancing or adjustment needs to be correct to get correct inference.
