The point they were trying to make regarding easyJet is that they would not pay for any missed onward connections from your arrival airport that weren’t part of your itinerary. No airline, KLM included, will leave themselves liable for missed connections that weren’t part of the deal for want of a better term.
EasyJet would provide hotel accommodation for any overnight delay or if that wasn’t possible, reimburse the cost of a hotel.
Regarding your delays on the CPH route, when you look at the flights delayed at an airline you often see it’s the same ones over and over again and often a result of air traffic restrictions that are the bug bear of European aviation. The plane operating that flight will operate the same rotations each week/day before operating the flight, experiencing the same pinch points. Not an excuse just a bit of background info.