A divemaster/dive guide has an employer, and the employer defines the role of the employee. In a thread a few years ago on this topic, several dive operation owners chimed in, and they all agreed that the safety of the divers was the dive guide's primary responsibility.
Monitoring the safety of the divers can look very differently depending upon the divers. A diver who demonstrates strong competence will be given a lot of freedom to lag behind, poke around, wander off a little, etc. That same guide giving divers such freedom may spend an entire dive literally holding a diver's hand, something I have seen more than once.
EDIT: I used to use a dive operation in Florida that announced that they would have a DM in the water during wreck dives, and if divers wished, they could tag along with the DM. Otherwise, divers were on their own. That was that employer's expectation for the DM.