ᐅ Bathroom exhaust fan instead of kitchen range hood?!

Hello,
so far, we have managed in our apartment for 40 years without a cooker hood or extractor fan in the kitchen, even though a lot of cooking and baking takes place.
However, we are now increasingly bothered by the humidity caused by cooking during the winter.
Unfortunately, there is no practical way to install a "regular" cooker hood above the stove (for example, an island hood or wall-mounted hood) — mounting a hood directly under the ceiling might still be possible, but that seems too complicated for me.
Now I am wondering what would happen if a standard bathroom exhaust fan were installed in the exterior wall at a height of 2.30 meters (7 feet 7 inches), about 15 cm (6 inches) to the left of the stove (for example, DN 150, 300 m³/h (176 CFM)).

I think without a grease filter the fan might eventually look dirty... but so far we haven’t had any grease problems in the apartment. Would streaks possibly form on the wallpaper near the fan? Or could grease even drip out of the fan on the inside or outside?
Has anyone done something like this before?

Thank you very much for your help!

** Among other reasons, because the back wall is 15 cm (6 inches) behind the stove and to the left of the stove, at head height, there is a window that opens inward.

That would cause streaks to form, and yes, grease would accumulate on the extractor fan. I can only advise against that. Without a range hood, it’s quite unpleasant; you would always have to keep the windows open when cooking or baking, which you should do anyway.

There are also very expensive retractable exhaust vents, which might help. I wouldn’t do it without an exhaust hood; the grease filter is extremely important, and without it, your fan will soon be full of grease and dirt.

There is also a downdraft extractor available (Bora cooktop extractor).