... why can't there be a private company that provides fire response services?
The competition from the public sector service destroys any potential profit margin for a putative private operator.
Think about it. To make it profitable, you'd have to offer something the public sector service does not, or do it more efficiently. There are many, many things wrong with many parts of the public sector, but the fire service actually does its job fairly well. It doesn't do it in the most efficient manner, but the marginal cost is low enough to most people and companies, that the benefit of paying for a private service is limited. Therefore profits are too low esp. given the risk/capital costs of providing the service.
If the public sector service didn't exist, there definitely would be profits to be made from a private fire response service, but given the dominant market position of the public sector, combined the in-built "advertising" spend they have, it would be very difficult for a private company to compete.
Now, in settings where the public sector service doesn't operate, there are opportunities for private provision. For instance, BAA/Ferrovial operates and pays for its own fire service within the UK airports that it owns. They're supported by the public sector service if needed, but otherwise they provide the emergency fire cover within the privately owned airport.