There is one thing bothering me for some time: What does area under Kaplan-Meier survival curve actally represent?

Is there a plausible way to compare two areas under curve when for instance two curves cross late in follow-up and invalidate the use of log-rank test due to violation of proportional hazards assumption? In example: treatment provides early benefit and we can see that the curve is higher than the control curve for substantial proportion of time but two curves later join and same survival rates can be observed afterwards. Same number of patients died but treated patients died a bit later. Can we compare follow-up times only and say that treatment adds "years of life" although it does not improve survival??

Can patient-time approach (if this would be the right name) substitute for log-rank test in this situation? Does the area under the curve represent months/years of life experienced?

I would be most thankful if someone could help me understand this better.