Up until now, I've always left my wort to cool naturally in the kettle after boiling. Depending upon ambient temperature, and the volume of wort, this can take quite a while (typically 24-36hrs). To minimise the risk of infection, I normally pull a new bin-bag over the top of the kettle & pull it tight, straight after the boil, to exclude air and also fruit flies etc.in the summer. I have never had a spoiled brew using this method (I really, really shouldn't say that, should I?!!)
I've only just acquired fridge for fermentation quite recently. So far, I've only used it as an insulated box, not as a fridge, employing heating only to keep the fermenting beer at temperature. I hadn't thought of using it to cool the wort, but thanks for the idea - it sounds like a decent plan to me.
Of course, I am in a hilly part of SW Scotland - so it rarely gets all that warm, never mind hot here. At the moment, I'm still using central heating and a woodburner to maintain 20°C in our living rooms morning and evenings. Unheated rooms are about 12-15°C at the moment (all day if on the shaded side of the house).