The problem is the sometimes exorbitantly expensive activated carbon filters. For our cooktop, it costs around 120 euros and should be replaced twice a year. I prefer exhaust air (where possible, of course).
For bargains, keep a reserve. Changing every 6 months is not excessive with heavy use, though. Significantly higher running costs cannot be avoided compared to exhaust air, unfortunately that's true.
We also have an island and the exhaust air goes outside through the floor! We have a 25cm floor structure including insulation and a 10 cm high flat duct simply disappears in the floor beneath the insulation and the screed!
Exhaust air means, however, an appropriate opening in the masonry. Maybe it’s not about what is technically possible, but that the facade comes from the architect and must not be altered. With this argument, a slight modification of a window was also denied to an acquaintance of mine.
It may also technically not be possible if you have [Kamin], [Kontrollierte-Wohnraumlüftung], and [Ablufthaube]? The chimney sweep told us that the pressure switch would trigger an alarm as soon as the [Ablufthaube] is in operation.