Exhaust fan instead of kitchen hood in the kitchen?!

Hello,
so far we have managed in the apartment for 40 years without a stove hood or cooker hood in the kitchen even though a lot of cooking and baking is done.
Now we are increasingly bothered by the moisture from cooking in winter.
Unfortunately, there is no possibility to properly install a "normal" cooker hood (e.g. island hood or wall hood) above the stove ** - a hood directly under the ceiling would still be possible: but that would probably be too complicated for me.
Now I am wondering what would happen if a normal bathroom fan were installed at 2.30m height in the outer wall (15cm to the left of the stove) (e.g. DN 150, 300m³/h).

I think without a grease filter the fan might eventually look unsightly ... but so far there have been no grease problems in the apartment ... would stripes possibly form on the wallpaper near the fan? ... or would grease even drip out of the fan inside or outside?
Who has done something like this before?

Thank you very much for your help!

** among other things because the back wall is 15cm behind the stove and to the left of the stove at head height there is a window that opens inward.

Stripes would form there and yes, grease would remain on the fan. I can only advise against that. Without an extractor hood, it's pretty bad, then you always have the windows open when cooking or baking, you should at least.

There are also very expensive retractable exhaust vents, that might help. I wouldn't do it without an exhaust hood, the grease filter is extremely important, and without it you'll soon have your fan full of grease and dirt.

downdraft ventilation is also available (Bora cooktop extractor)