Kaplan-Meier curves have the advantages and disadvantages of displaying simple versions of raw data for members of groups. Someone with knowledge of the subject matter can gauge whether the data sample is representative of the population of interest. Survival differences among groups are easy to see. If survival curves for different groups cross, you can consider what that means in terms of the situation you are studying.
Unless the data are from a randomized trial or the groups are balanced for other outcome-associated variables, however, the Kaplan-Meier plots will hide the potential contributions of other variables to outcome. If those variables are also associated with group membership, you might be misinterpreting the apparent survival differences as being primarily due to group membership per se rather than to the group differences in values of other variables.
As you note, the log-rank test is only for evaluating whether there are any differences in survival curves among the groups. It also doesn't take other variables readily into account, as with additional categorical predictors you end up with multiple small groups and continuous predictors would have to be binned into groups. Frank Harrell thinks that the log-rank test shouldn't even be taught as a separate test any more, as it's just a special case of a Cox proportional hazards model. A Cox model can deal directly with multiple and continuous covariates.
I suppose that it might be possible for a log-rank test to show no difference in overall survival curves while median survivals are different, but (1) it's hard for me to think of such a situation and (2) you would then have to try to understand how to explain such a situation based on your understanding of the subject matter.
With non- or semi-parametric survival analysis based on Kaplan-Meier or Cox models, there are important differences between comparing survival probabilities at a given time and comparing median survival times. The primary variance estimates are related to survival probabilities at given times. It's straightforward to test for significantly different survival between two groups at a given time by comparing the survival difference against the corresponding standard error of the difference. This page has a simple illustration.
Group comparisons among median survival times in such models is trickier. For a median-survival-time confidence interval, you find the time points at which 50% survival intersects the corresponding upper and lower confidence intervals of the survival estimate as a function of time. The errors in median survival times, unlike (properly transformed) survival probability estimates, don't typically have near-normal distributions. That complicates median-survival comparisons among groups. This paper reviews some methods and proposes a new one to get around the problems: Chen, Z., Zhang, G. Comparing survival curves based on medians. BMC Med Res Methodol 16, 33 (2016). I don't have experience with that method, however.