Why penalize the head coach? Officials on the field had no idea what was going on, all they knew was the coach essentially got attacked, they had their radios but those are different frequencies/channels than the coaches' ones, for obvious reasons. Here, clock operator is not part of the crew (sites provide them.)
Therein lies your problem. Ever so fortunately, WAY back (prior to late 1960s), when Game Clocks were the latest fad, some very wise Officials, locally, decided the responsibility of game timing was an Official responsibility whether it was handled on the field, or running a scoreboard clock. Despite some considerable initial objections and resistance, they prevailed and eliminated a gaggle of various problems and thus far, have avoided similar objections with the emergence of the latest "necessity", the emergence of scoreboard "Play clocks", which are currently also handled, exclusively, by game officials (either on the field or in the booth).
Well trained, IMPARTIAL, consistent, Official clock management & supervision of Game time (either on the field, or in the booth) has proven to avoid all sorts of unnecessary arguments and controversy at EVERY level of the H.S football experience and proven to be well worth the minimal added expense. It provides effective, IMPARTIAL and consistent event time management, far more easily adaptable to rule adjustments.