Does anyone here in the forum possibly have a recirculation extractor hood combined with a separate active ventilation system in the kitchen in use?
Yes, we do: recirculation extractor hood + exhaust valve of the central ventilation system in the kitchen. The living room gets fresh supply air through its own valves, from there it flows into the kitchen and leaves there as exhaust air through the valve. It’s great because moisture and odors are (slowly) removed—without window ventilation.
Different technology but comparable in terms of comfort.
How important is the extra ventilation system from your point of view and would the benefit justify the cost of 800 EUR?
You can only answer the question about comfort yourself. The kitchen needs fresh air, either through a fan or through a window.
Beyond that, the question arises how the fan is integrated into the ventilation concept of the entire apartment and into the energy calculation. You shouldn’t just be able to leave out a part. So where does the supply air come from?
Noise level: the fan is probably audible. But if the open kitchen does not border directly on the sofa and you don’t live next to a noise source, then the fan should not be disturbing, I think.
By the way, there are also devices with heat recovery, but that would then be in recirculation mode (see ventilation concept).