Long ago, when electronic scoreboards that included timing devices became the fashion, our Chapter decided that such timing devices would only be allowable when they were operated by member Officials. Thankfully that policy has been accepted, although it has been seriously debated a number of times over the years, but remains policy. During several of those debates, the issue of adding the responsibility of maintaining the other scoreboard functions (down, distance, TOs, etc.) to the assigned Officials has come up, and has been rejected, and has been designated as the responsibility of the Home management, which has been working well, for decades.
The policy hold for all levels, including multiple "Youth Football Leagues" who had also previously appealed, which included a long ago trial of maintaining an "official game clock" on the field by the field Officials and allowing a scoreboard clock to be maintained (for the benefit of fans and Teams) by Home Management. Not surprisingly, that turned out to be a disaster, causing far more confusion and problems, and has since been discontinued and remains prohibited.
We remain always thankful to the wisdom and resolve executed by those former members resolving this issue many years ago.
Officials assigned to clock operation are fully dressed, as the other Field Officials, are subject to the instructions and decisions of the Referee and enter and leave the field with their crews.