Also on Alltrails is the path to Agia Marina, which I get the impression is the one they think he may have taken? The reviews say it’s rough walking.
The hiking map also shows this path, and then one from Agia Marina to Symi Town running further north.
It seems a very strange decision to try and go this way, especially as they would have seen this area from the boat out to the beach in the morning. But, people do all kinds of irrational things, especially if feeling unwell or suffering from heat stress.