So many helpful answers thanks, if I use them all my wifes going to think I'm making excuses and just want her to suffer in the kitchen heat!
One thing that crossed my mind is: could a ceiling fan be dangerous in the event of a fat/oiil fire - fan the flames?
Maybe this another reason to add to the list of not having one?
I would suspect the answer to your question above is Yes. I used to run Restaurants: Food Hygiene Regs in the UK - have an understandable rationale - rare with Bureaucrats! You wife is suffering the same sydrome as all Chefs - even in the UK during the Winter! This is the reason WHY no Fans or Air Con - and I shall not bore you with temperature control and separate fridges, chopping boards, detergents, handwashing, etc, etc. Although, I try, where practical to follow in a domestic situation.
ANY cooling aparatus in a Kitchen will allow condensation to form on top of the food. Condensation - however minimal allows bacteria to grow very quickly on the outside of the food - which is why warm or hot food must never be put immediately into a Fridge after cooking (cover and lay to one side). Cooking and Heating Food properly kills any bacteria which may have formed after cooking in residual cooling in the Fridge.
What can help alot is an very good Extractor Fan (ducted to an outwide Wall) if placed above the Cooker - particularly as far as steam, frying, open grilling is concerned. Test efficiency by holding a piece of paper loosely near the screen of the Fan - if the paper flys out of your hand and sticks to the screen - it is a good Fan! Ducting and Fan should be cleaned thoroughly at least once a year as fat, grease, debris builds up and efficacy decreases (we did it every 4 months in the Restaurant Kitchen). Opening the Window can help too - but you would then need removeable Fly Screens: We had a curtain of fine chains - so the window could be opened from the inside.
Hope this helps - and the condensastion issue is "killer blow"

.....I am sure you wife would be averse to bugs and bacteria multiplying on your food as you would be

Best wishes