Let water run from one gutter to another - search for photo

Here is at least a picture of the entire side of the house, in case that helps. (Somewhat older, therefore without this structure.) You can also see that the drainage originally ran through the house, that was changed.
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that it is initially only the emergency solution.

No, no, this is the final solution. The inlet funnel etc. is already all attached, and since it is soldered, that is also how it is intended.

already attached

It was already attached before a solution was discussed... and it is not a big problem to replace it.

Well, there is an execution plan where the gutters and downpipes are usually marked.

And yes, of course, it is no problem to exchange something. But exchanging this, exchanging that, and the workday is over. The roofer has to be willing to cooperate as well.

Remount the gutter so that it has a slope to both sides of the roof. Lead the drain from the gutter into a short pipe without a bulky funnel, which goes over the first wave of the tiles and leads to the lower house gutter. Visually, this is better. In heavy rain, an overload can occur - so what, nothing breaks anyway.

The second image is still heavily obscured by the scaffolding. Where does the drain run?

How about something like this: