Sorry for the late follow-up. The modelled cowl flaps were always open, they didn't budge at all, and they were matching what's visible on the tablet -> control issue.
While the audio cues for the electric motor gave me a false sense of working controls, FSUIPC interfered for SOME reason. I didn't have cowl flaps assigned to any hardware controls, but FSUIPC inherited the calibration settings for cowl flaps axes from another profile. When I turned that off (ie. when I reset the calibration settings for the unassigned axes), the cowl flaps started working.
Thank you, Nicotine, for pointing me in the right direction with the images of the open and closed cowl flaps. I never checked their state, for some reason I had this idea that the cowl flaps were internal and I hardly pay much attention to the outside of my planes unless the dev forces me to do a walkaround. I learned that I can actually observe the left-side cowl flap from the pilot seat.