TDs sure. At Winthrop, there were slow open groups with two+ hole gaps in front of them. If the course director, they had 3 I believe, each responsible for 6 holes, put the group on the clock, the group would be given reason to call a slow player taking excessively long.
Volunteers should never have such a power.
Not entirely true. Course marshals (not directors) are empowered to make calls by the rule book (specifically Competition Manual section 1.11 D), they are only discouraged from actively making calls on things that affect only individual players within a group (like say stance violations) and can be construed as unfairly targeting one group/player over another.
However, when it comes to keeping the flow of traffic moving, so to speak, I think they are already empowered to do what is necessary within the confines of the rule book and Competition Manual. The pace of play rule in the Competition Manual (section 3.2) says that "[a]ll competitors shall play without undue delay and will make every effort to keep up with the group in front of them". If a group has two (or more) open holes in front of them, and they're not the lead card starting 30 minutes after everyone else, then who else but a course marshal is in position to speed them up?