So, I experienced a new and a further development along the same lines during this weekend's trip.

I was able to check in as normal on the app...but the app could not/would not display the boarding pass.

Instead, it would take me to a screen which promised to send a "check-in confirmation" to my email address - and despite verifying the pre-filled email they displayed was correct, and hitting "send" multiple times, no such "check-in confirmation" was delivered.

Now, I was able to retrieve the boarding pass from the website, and to print out boarding passes at kiosks in both airports - but still, this is the first time the KLM app has failed to be able to display the boarding pass.