Repair successfully completed; the heating system is working perfectly.
Image 1 shows the inside of the heating system. Image 2 shows the old control unit from the inside. The mass has the consistency of gelatin and already has cracks. The cables on the connector also don't look good anymore (Image 3). I'm very glad that I replaced this part. Thanks to Mulle for the support in this matter.
Image 4 shows the new part. The adhesive material is black this time and harder than it was in the old version. It's also noteworthy how it's distributed: Hella apparently manufactures this part and supplies it to Eberspächer, who then supply it to vehicle manufacturers, who in turn sell it to end consumers or workshops. If it's sold as an original part, three companies have already made a profit.

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@Dieselbear: Now you know why those things are so expensive.
Image 5 shows the new motor. If anyone needs to replace this motor: The actual manufacturer of this motor is not Eberspächer, but Bühler in Nuremberg. This motor has the part number 1.13.055.030 and costs around 30 euros from Bühler. From the heater manufacturer, it's sold as a combustion fan for 100 euros or more. There's also the black plastic housing attached, which you can reuse from the old motor. You can disconnect the two cables and resolder them, remove the cover, cut three M3 threads into the existing holes, put the cover back on, screw in the motor – and it's done. A significant saving.
I also replaced the flame arrestor screen. Surprisingly, it looked quite good. It was installed in February 2009. But since the heater is already being removed, I'm going to proactively eliminate this potential source of problems as well.
I ran the heater for one cycle and there were no error messages. The old motor didn't have any major issues, and the internal resistance was 2.3 ohms.
The heating system is much quieter now; it was definitely broken.