Internal insulation because of mold stains?

The apartment is to be rented as a shared flat. All fans run via a central control unit. I can attach that outside the apartment ^^

"Sorry, I don't know why the fans always break."

As I said, we manage it well, but we also do a proper shock ventilation 2-3 times a day. I can't assume that a tenant does the same.

Unfortunately, that is the risk you have as a landlord. Only a reasonable cooperation helps. As a landlord, you (unfortunately) have very few rights.
And Tassimat has simultaneously written what can happen. You are dealing with the tenant with the central control unit as if you enter their apartment 5 times a day to air it out. And that also with your own apartment key, which you are not allowed to have.

That wasn’t meant entirely seriously about the control unit being outside the apartment

And yes, a reasonable coexistence is the most important thing.

But if a ventilation system is present, then the chance of mold and trouble is lower. And if there is mold anyway, then you can check if the system is running.

With a system, the likelihood that everything will go well is quite high, I’d say. Without a system, mold will almost certainly develop.

Or am I wrong about that? If yes, what could I do better?

Do you have examples of such a [Wohnraumlüftungsanlage]? I will probably have to install it in every room, right? How is the noise level of such a system, and doesn't heat get lost again if I have a hole to the outside?

I have focused on the system from Fa. Südwind.
It is advantageous in every "relevant" room.
I install it everywhere except in the pantry and in the hallway with dining area.
The noise level is stated in the data sheet, but from my own experience I cannot say anything about it.
The system has quite a high heat recovery. So I think in the end not much more heat will be lost than if I ventilate normally.

... Südwind ... The system has a fairly high heat recovery. So I think in the end not much more heat will be lost than if I ventilate normally.

Estimated rather less heat loss than "normal ventilation". According to Südwind 93% => ~7% loss (as well as probably similar controlled residential ventilation with cross heat exchanger)

But is the wall construction really "30cm clay wall" and nothing else? (except plaster outside and wallpaper/paint inside) Year of construction?

I myself have a not so young house (~270 years) with various old materials. I have learned that the house must not be too airtight. Otherwise, there will be building damage due to moisture/water.

The shift of the dew point with interior insulation was already mentioned. That would be very dangerous to execute incorrectly!

Presumably, a partial renovation (windows?) has already taken place here that caused the problem in the first place? Here: ask a professional, renovate accordingly. Half measures usually don’t pay off. Also have the tax aspect looked at (ask a professional again).

Maybe it is worth it after all?