It’s not a terrible idea…but it’s also described as ‘windy’, so even if his umbrella or hat are found, he still could be some distance away from it.
It’s looking really bad now. It would be a miracle now if he’s found alive. With only 10 searchers out a time and limit hours due to the heat, if he’s somewhere that’s not visible they have very little chance of finding him. On this terrain they could walk right past him and if he’s in a crevice they won’t see. They can’t have movement and visibility to cross of sections as completely clear.
If they tried to have more people searching that’s technically difficult. They need a second team at base at all times to act as a contact point and be ready to rescue the rescuers if they call for help or go out of contact. They can’t let anyone who isn’t fully trained help as it’s too much risk, and they can’t drastically increase their numbers, even if the family wanted to pay for the staff…because they’d have to be brought in by boat/helicopter from various places, be fully managed safety by an expanded ground team and all these extra staff would need accommodation on an island that doesn’t have any large hotels…and most of its air bnbs will already be occupied during tourist season.
If they did find Dr Mosley alive, he’s clearly going to be in a near death state and possibly also gravely injured…they then have to get him off the mountain and transferred to a hospital capable of treating him…which could also be a process that is time-consuming.
Most people can survive up to 3 days without water, 5 days in unusual cases and 7 if you are being very positive for an extremely unusual case. In intense heat it can be less than 24 hours. From the description of the rescue mission it sounds like it’s fairly high risk for the staff involved, it’s not something they are going to continue with once they decide chance of life is at zero, however they may carry on for ‘show’ for longer than normal because they don’t want media criticism.
The water and drone searches they could continue for several weeks with a small scale crew in hope for a body recovery.
This island is my idea of a nightmare for a holiday. It’s all steep hills and difficult accessibility, with very little medical facilities and no airport. Even things that generally easy to manage such as breaking an ankle or getting covid in this place could end up as a massive ordeal or deadly. If you’re on medication and lose it, maybe you can’t get it replaced, if a phone mast goes down that could be the whole place cut off.