Hello everyone,
I wanted to give you a final update.
Unfortunately, the large VW partner network here in the city insisted that they would only work on the issue if I brought the car in with a built-in auxiliary heater, despite my having my own VCDS log and pointing out that they could either repair the removed auxiliary heater themselves or send it internally for repair. Just for the diagnosis, they would have charged 220 euros! Furthermore, I suspect that they would have simply replaced the part rather than repairing it, which would have likely cost around 1300 euros!
I then contacted Mr. K. in Regensburg by phone, and after several attempts, he offered to repair it for 250 euros, including a 12-month warranty, if I sent it to him. I accepted this offer and then, together with a friend, I started the disassembly. The disassembly took many hours because we had never done it before. The fuel line actually only needs to be sealed for the sake of cleanliness, but diesel doesn't leak out, contrary to what I feared. It was a bit tricky with the two water hoses because the two ends are different sizes, meaning we had a hard time getting the larger end onto the 19 mm connector. To my surprise, the auxiliary heater was from 2011, and since my car is from 2005, someone had already worked on it before. Finally, we refilled the coolant (including bleeding the system) because the level had dropped below the minimum.
I ordered some parts, such as the exhaust muffler, the air intake silencer, the air intake manifold, and the flexible pipe, from VW because the existing ones showed signs of increased wear and tear, and the flexible pipe was also damaged.
Attention: The air intake manifold for the air intake silencer was part number 1K0 819 547 A. I mention this because I initially mistakenly purchased air intake manifold 1K0 819 547. Strangely, in the AKTE diagram, which shows the auxiliary heater as a complete unit along with all accessories (exhaust silencer, air intake silencer, etc.), the situation is that for the air intake silencer 1K0 819 194 B, the air intake manifold 1K0 819 547 is listed as compatible. However, the actual air intake silencer looks different from what is shown in the diagram (namely, like the one originally installed in my car), while the air intake manifold only fits the air intake silencer shown in the diagram, but not the part with the associated part number (or my part). In the AKTE diagram, which only shows the auxiliary heater in the form of all its orderable individual parts, air intake manifold 1K0 819 547 A is listed, which also fits the air intake silencer and the installed bracket. Therefore, when purchasing, be sure to check that the delivered parts match the ones already installed!
After a few days, I received the repaired auxiliary heater, including a printout of the test stand report. A few days later, it was time for the installation, and when everything was installed, the big moment arrived: pressing the button on the climate control panel!

From that point on, the auxiliary heater ran perfectly for the programmed time in the car's computer, and the joy was immense!
After doing some more research on the topic, I have established the following rules for myself:
-Run it continuously for at least 20 minutes to ensure that the combustion chamber reaches a high enough temperature to prevent any carbon buildup.
- If the distance is too short - which rarely happens in my case - and I'm not using the auxiliary heater anyway, I press the ECON button before starting the engine to prevent the pre-heater from turning on. If the auxiliary heater is already running, the ECON button doesn't make a difference.
"Potentially damaging" to the auxiliary heater is likely its operation as a pre-heater. It activates, in particular, when the engine speed is > 300 rpm, the coolant temperature is < 69°C, and the outside temperature is < 5°C. For example, if the coolant temperature is 50°C and the engine is started, the pre-heater will likely activate for only a few minutes. If this happens frequently, without the auxiliary heater occasionally running for, say, half an hour, it is likely to fail prematurely. Therefore, always remember to use the ECON button and allow it to be used as appropriate for the situation.
Okay, I'm glad everything worked out, and I hope this thread is helpful to anyone who's experiencing the same issue.
Best regards,
Dieter.