The cable isn't likely to be the limiting feature here. The method of energy conversion doesn't just require heat, it requires a
temperature gradient across the device. Here they're putting the tip in the fire to heat it, and the little bowl at the back full of water is used to cool the other end, thus getting a good temperature difference.
In contrast, throwing hte whole thing into the fire might work briefly while the heating was uneven, and the water hadn't all boiled off, but as soon as it reaches the same temp all over, no more power. And if it's all in the fire, even the water evaporative cooling isn't going to help much, vs when it's out of the direct flame/hot air stream at the edge.
I imagine you could design a in-fire version using a heatpipe to transfer energy out to a radiator/cooler somewhere, but that would be much more fragile and/or expensive.
What we really need is MEMS (or at least, really teeny, zero-maintenance) Stirling engines :)