Self-answer: as suggested by @Allan, if in any doubt, and especially if you still are in the warranty time, the best choice would be to contact your dealer.
Apple suggests to clean the keyboard with compressed air. I did that to no avail, and also gave the whole keyboard a tender scrubbing with a (dedicated, clean) toothbrush. This didn't help either.
On the next day, the key decided to get completely stuck in the "pressed" position, and I simply had no time to go to my dealer. I decided to gently lift it with the help of a thin, sharp tool. The lower part of the key (the split cursor up/down key) came off just fine. Below that there was the usual butterfly construction. It is delicate; there is a tiny strip of plastic which is not really attached to anything; don't blow on it or it will fly away.
I gave it a very tiny airstream to get away any specks of dust, and just put the key on again. Lo and behold, it is now unstuck, and also my original problem of not registering the keypress in 50% of activations is gone.
If you google, you quickly find videos about how to disassemble and clean the butterfly mechanism itself, but I did not need to do that. The "half" key looks just like the regular keys, just literally split in half; it is not any more complicated.