An idea as I sit by the fire

Technically....yes, but in reality the energy being transferred by the "air up currents" is "meager" at best. Probably of the order of single watts. To put that into context if your fan generated 1 watt, you'd need 2000 fires to boil a kettle!

However trapping the heat directly that's wasted going up the chimney and heating water is a very good source of free energy. We don't seem to do this much anymore as we've transitioned away from open fireplaces in homes, but the traditional "back boiler" of a fireplace in the 60's / 70's was pretty good way in heating the hot water.

This. The device is utilizing Peltier effect. You can buy peltier element separately. I had device made and used CPU heat sink and fan. Placed it on top of yankee candle jar and it worked. The thing is you need a fan to cool the cooler as otherwise the difference is not big enough. So although it does generate small amounts of power you will use that power for fan. Unless you can achieve efficient passive cooling.

As someone mentioned, they use peltier modules - they can be used to create fridge/cool conditions too. I have had a few over the years and one got very noisy so I took it apart and tested it on the stove. With the heat from my stove I could certainly have charged a phone. I was seeing slightly over 5V and up to 1amp of current. IIRC I put it on a board with a tp4056 too and it managed to charge a 18650 fine - over a good few hours. The biggest problem was the heat on the cables!!

Herself wasn't too happy with the look of it so I didn't get past 1 or 2 nights testing

Hah - I had one a few years back too - motorised for the full arc. Lasted a few months too before there was uproar! Ah well....