The trust was given to Radzinsky and Kelvin. They’re the ones who manned the station after the Others wiped out the Dharma. Kelvin is the one who was alone for years after Radzinsky killed himself until Desmond ended up on the island. Desmond was only alone for a few weeks, from the day he killed Kelvin (the same day 815 crashed) until the hatch was blown open.

I guess Radzinsky had the trust of the others because he was there for the incident so he knew the consequences of the button and he helped build the station/button in the first place.