This seems like an interesting short survey of London firefighting history, but I'm missing the debunking people are remarking about. The only thing that slightly resembles debunking is that in the conclusion the idea that fire brigades ever let houses burn is referred to in passing as a "legend."
In regard to the legend - I don't think that anyone ever said that fire brigades would refuse to put out uninsured buildings where the fire might spread to other buildings, that fire brigades couldn't be induced by cash to put out the fires of the uninsured, or that cities never offered baseline inducements for fire brigades to put out fires whether the buildings were insured or not. That would be adding content to the "legend" in order to debunk it. Looking at the record and finding out that all of these things happened is expected and unremarkable.
The fires that we would expect not to be put out if the "legend" were accurate would be fires that happened to uninsured buildings that didn't endanger nearby structures, buildings owned by people who didn't have enough cash or credit to convince an uncontracted fire brigade to act, that were close enough to the city to be accessible to a fire brigade yet far enough away that the city didn't feel responsible for them. Additionally, the chance that a building owner might simply pay (under flaming duress) a brigade that showed up would mean that they probably generally would show up, and if they weren't ever willing to watch a building burn, they'd be undercutting their own last minute deals.
These are the exactly the fires that we would expect not to be in the record. Fires in detached structures, owned by people with no money.
It seems that the city started rewarding brigades for putting out fires in uninsured households based on the order in which they arrived. Additionally, they were consolidating/federating in order to take on larger contracts - which incidentally meant [original scholarship] that the consolidated/federated brigades would always get the first arrival bonus through this collusion. As the number of independent brigades dwindled and the socialist city reward money became large enough that the actual insurance element receded to only being profitable for large contracts rather than individual homes, the consolidated block of fire brigades organized their own nationalization.