While it's broadly accurate that private fire brigades were highly competitive and staunchly capitalist (and exploitative in nature), the idea that any of them would refuse to put out a fire without first being paid has been debunked.
It's not impossible that it happened on some occasion, but in general this would have been a horrifyingly terrible business decision: not only would they be lambasted by their competitors, but they usually covered at least some of the neighbors of any given burning property. It would have made their jobs infinitely more difficult when it came time to make good on protecting their clients if they let nearby fires grow (and grow they would, very quickly).
See here for a retraction of a previous piece by Tom Scott on this topic as it relates to the UK, where these stories originated, including reference to a correction of his original source, the London Fire Brigade Museum's webpage. https://youtu.be/Wif1EAgEQKI All of the points made in that video about the UK apply the same to the US, as far as I'm aware
What DID actually happen was that brigades would send someone to prevent their competitors from putting out the fire before they could arrive (e.g. by blocking access to a hydrant/water main) in an effort to guarantee they would get the payout for putting it out. Similarly shitty, for sure, but not quite the same
As I said, there is I'm not surprisedthat it has happened throughout history, and that's super fucked, but even that article has other firefighters going "WTF is wrong with you? Why would you do that?"