Many airlines say that unless told otherwise that you check in based on the time of the original flight time and not the delayed time. This especially applies when an airline amay only have one or two flights a day.

Sure you may have an airport wait but stuff happens.The flight could land earlier and they don't do a full clean to try and get away earlier. But unless specifically told always check in based on the original flight time. And yes that includes luggage.

Just because CPH staff weren't DL employes makes no difference. They work to DL policies just as directly employed DL staff do. The 'supervisor' and 'manager' could have been a DL employee at the end of the phone somewere in the US not just in an office at CPH

Not outside the US but I recently had a BA flight from LAX that was delayed by several hours. BA's website is something less than clear in this regard but I received a text from BA that said the flight "is delayed and will now depart at 23:40 local time. The check in desks will close at 22:40. BA apologises for the delay."

Original flight time was 21:35. We showed up (with bags) at about 21:00. The check-in desks were in the process of converting over (at LAX in the TBIT they're rotated among various airlines) and the BA signage had been removed, but I knew where they were supposed to be and a harried BA agent did check the bags in. There were a few other people checking in and the whole BA operation was being moved down the wall to make room for the next airline up.

If we had shown up an hour later - as in a good 40 minutes before BA told us the desks would close - I have no idea what would have happened. Similar situation to yours in that there was absolutely zero chance of the incoming metal showing up on time since I could see where it was, somewhere over Arizona. And we'd been told specifically when the desks would close.

Odd, considering the frequency with which flights are delayed, that the process is so random.