I never was an EME officer, but, given that no one else will answer and I have a few friends who were ...
EME officers are, primarily, leaders and managers. As you noted there are well trained, highly skilled specialists in the EME branch who do all the many and varied jobs. In the field, in combat arms units and in whatever we now call the field workshop (maintenance company?) the officers have to lead: including in ensuring that the infantry battalion's administrative echelon is defended, moved, concealed, etc; and manage: ensuring that the technical maintenance tasks are done, on time and right, the first time, because the rifle companies need their weapons and vehicles and radars, etc.
EME officers rise up through the ranks â â€œ commanding units at combat service support platoon, company and battalion level and they frequently fill senior staff positions at the colonel and brigadier level â â€œ some, like Bob Fisher, are exceptional men by any standards and retire as lieutenant generals, ranked by many, many of their peers as being amongst the best CDS we never had.
EME officers also serve on bases â â€œ including air bases and, maybe, even the dockyards â â€œ supervising the maintenance of 'land' equipment which includes e.g. the fire trucks and runway clearing equipment on operational flying stations.
EME officers, like supply and transportation officers are vital members of the fighting forces â â€œ land and air forces, too.
EME officers also staff the equipment engineering/procurement staff in Ottawa â â€œ deciding what equipment meets the specifications required by the people in the field and then getting it on time and within budget â â€œ despite a steady torrent of military, bureaucratic and political interference in the process.