Crazy busy this week. I finally stopped by the library to look at a globe myself. It looks to me like you can do it. My string wasn’t long enough to go all the way, but it looks to me like you can get from Pakistan through the straits between Antarctica and S. America. I’m not sure why you landed on Vinson Massif.
I suspect the problem may be that it is a straight line but does not lie on a “great circle”, i.e., a path that traces the full circumference of the sphere. Think of latitude lines: they are all “straight lines” (as that term applies to this discussion), but except for the equator they do not trace the full circumference of the earth. Now think of trying to draw a perfect circle on the globe that is not a great circle and does not run perfectly east-west or north-south. Someone watching you might not think you were drawing a “straight line”. Same thing with using string. If you saw me running the string from Pakistan past Madagascar and Cape Horn, you might think I was cheating by allowing the path to curve when in fact it was straight.
But I could show that I was making a straight line by taking a loop of string that was the right size and laying it on the globe in such a way that it made a circle, not an ellipse, and holding the globe in such a way that the circle was parallel to the floor, so that you could look down on it from above and see that it was in fact a circle.
Now, if we can do that in a way that puts Pakistan and Kamchatka on the circle without touching any other land in between them (along the arc that connects them through the southern hemisphere, of course), then that will show that there is in fact a straight-line water route that connects them.
You can also get your mind around it by imagining a flat plane intersecting the Earth in such a way that makes such a circle.
(If you tried to sail it you’d probably be sunk by drifting icebergs, but that’s beside the point.)