Magura rim brake does not work well on a new rim
As it turned out, the problem were not the brake pads. The issue was cause by the fact that the new rim is slightly  narrower. The brake cylinders still touched the rim, but were the already nearly fully extended out. This lead to the particular situation that the brake started breaking lightly, but full force coupled not build up, because the two cylinders had reached their maximum extension before that.

The fix was easy: I moved the brake cylinders (in the released position) further in and now they are gripping the rim again properly.