I fixed the shifter today. When I removed it from the car, I noticed the gate plate was placed in the right direction, but the "parallelogram pin" was not contacting the plate at the right spots.
I put a pin to line up all 3 arms, this was all the way to the right, ready to go in 5th (or secret reverse in my case...):
This was all the way to the left, shifter pulled upwards. Can't go further than the 1-2 gate:
So carefully removed the shifter itself from the housing, making sure none of the linkages and the spring washers get out of alignment. I loosened the 2 torx screws holding the parallelogram pin thing to the shaft and rotated the pin in relation to the shaft. I put it back together and everything works as supposed to.
Tomorrow I'll start pulling the transmission to check the clutch.