Hotel Tango, you are more or less right; at many line stations with most airlines there would have been at least a Station and/or District Manager, in one guise or another. But, except at their home base, most airlines purchased their ground handling from another airline (especially if pooled) or an agent.
But that's by the by; what counts is the airline's determination, or lack of it, to ensure that their customers get the best possible service, which means, in the case of delayed flights and missed connections, starting work on alternatives as soon as it becomes obvious that the problem is going to happen, not just doing SFA and reacting pathetically badly when confronted with a large group of tired, unhappy and distressed passengers becoming, understandably, increasingly irate at the connection airport. Who does this depends on how the airline runs things, but what the airline cannot escape is total responsibiity for making sure they or their contractors do it and do it well. Some manage this, as Paulc reminds us.
As for the case that triggered this thread, let us remember that this was Emirates/DNATA in action at their own home base! Thus no other parties involved. If they screw up so badly there, it takes little imagination to foresee how badly they will screw up when their passengers are in the hands of an agent where there is no Emirates representation.
Let's also remember that the agent's eagerness to go the extra mile might be tempered by Emirates'/DNATA's payment policy (yes, bitter experience!) to delay payments for months, then argue the toss on spurious grounds as a delaying tactic, and finally do nothing in the hope that the payee will give up. Perhaps their accountant trained with Ryanair.
Perhaps a good old fashioned phone call to Emirates Support would have helped.
It's why I asked if anyone could supply the number of a Customer Service manager. No-one did, perhaps because it's a closely guarded secret. You would need to believe in fairies to believe that simply calling Emirates Support call-centre, from a pen in DXB airside transit area, if you can find it's UAE number (wi-fi was allegedly not working properly), to ask for decent service now would achieve anything. (It's quite indicative that of the 4 topics they list for telephone support, each one is about something Emirates want to sell to you!)