Scenario 2 can't in general be counted on to demonstrate a survival difference between drugs A and B.
Say that with drug A 1/3 of patients die before 800 days, 1/3 die between 800 and 1200 days (with half of those before and half after 1000 days), and 1/3 die between 1800 and 2200 days.
Say that with drug B 1/3 of patients die between 800 and 1200 days, 1/3 die between 1800 and 2200 days (with half of those before and half after 2000 days), and 1/3 die after 2200 days.
Then you have the specified median survivals of 1000 days for A and 2000 days for B. Median survival is clearly better for B.
Nevertheless, 1/3 of both groups die between 800 and 1200 days, and 1/3 of both groups die between 1800 and 2200 days. Sampling scenario 2 would show no difference in deaths between drugs A and B despite the longer median survival with drug B.
Yes, the above example might be extreme. You don't, however, want study design to depend on untestable assumptions about the shapes of survival curves, hidden assumptions that underlie the proposal of sampling scenario 2.
With scenario 1 you directly get information about the shape of the survival curves. That would be important for testing the proportional-hazards hypothesis that underlies the Cox regression proposed in an earlier version of this question. (Note that for a simple 2-treatment comparison you could use a log-rank test in scenario 1 and avoid such assumptions.)
Scenario 1 also gets your results faster. This calculator for survival studies with 2 equal groups at the start shows that, under your assumptions (200 patients in total with data, all starting at day 0, 1000-day median time to failure with drug A, a hazard ratio of 2) you will have 80% power to detect (at p < 0.05, log-rank test) a survival difference after 850 days of follow up. There will only be 46 deaths total at 850 days. You will barely have started collecting data under Scenario 2 at that time.
It's frankly hard to think of a situation in which scenario 2 would be preferable, although that might just indicate the limits of my imagination. For example, under your hypotheses with exponential survival curves, I calculate that 59% of the deaths between 800 and 1200 days would be with drug A, while 51% between 1800 and 2200 would still be with drug A. If you wanted, say, to compare those proportions between the 2 time intervals as a test of differences in survival, Russ Lenth's power calculator indicates that you would have less than 20% power to detect that difference in A/B proportions with only 100 deaths sampled within each time interval.
If you would like to explore different sampling times like sampling scenario 2, the following plot shows the fraction of all deaths that occur in patients who took drug B, as a function time into the study, within 400-day windows, with exponential survival times.
Remember that scenario 1 gets you useful results by 850 days. That takes advantage of the low proportion of drug B deaths at early times so it's easiest and fastest to see differences between A and B at early times. As the sampling time gets later the difference in deaths between drugs A and B in the types of time windows you propose gets smaller (making it harder to detect B/A differences) until they are about equal at 2000 days, after which increasingly more deaths are associated with drug B and B/A differences become easier to detect. Again, it's hard to see what that type of sampling approach would provide over the standard scenario, scenario 1.