BOAC,
I entirely agree, it would have to be military radars doing the finding, and silly o'clock on a weekend is probably not the best time to keep a keen look out.
Nevertheless, the timings of the Inmarsat round-trips should give a range of plausible tracks north too, and would help anyone to run back through saved data to look out in hindsight. The problem with the Southern track is there's absolutely nothing to run into.
Mr 172 has noted the plausible possibility of cellphone sign ins over more populated areas, and I suspect the lack of those should be able to rule out a northern route, making a trip south with no cell towers until Patagonia the default.