Is a drip edge mandatory?

Do I understand correctly that you assume that a drip edge is not absolutely necessary?

You can call it whatever you want, and Fritz may think something else is suitable than Franz. Some suitable protection is needed to prevent water from causing the wood to rot. Construction fashions may change, but the necessity itself has always been and remains timeless. And if I understood correctly, your core concern was not, like Shakespeare, about metal or no metal, but the justified fear for the lasting integrity of the fascia board.

Does the water running off the underlay membrane run down the inside of the fascia board? Or does it go around the top over the metal sheet(?) into the gutter?

The said metal sheet only starts above the counter battens. If implemented this way, at least the water does not accumulate at the fascia board. If the roof is properly covered, hardly any water is to be expected on the underlay membrane either. Possibly condensation or wind-driven snow. Of course, structural wood protection is preferable, but I think in this case a properly treated board should last many years.

Has anything else come up on the topic or did you leave it at that?

Has there been any development on this topic or did you leave it as it is?

After multiple inquiries, the site manager told us: "If we knew everything better, maybe we should have built a house ourselves." We will raise concerns at the acceptance and if defects occur afterwards, we will stand our ground again. Actually, this is not right, but we have no desire to pursue it any further because we only encounter ignorance and no signs of compromise.