This would not really work because, for the most part, fire protection is a public good. It is not really possible to provide it to one household without providing it to others.
It might be possible if all the houses were single-family houses and if they were so far apart that a fire from one could not possibly spread to another.
But in a city, it would be impossible. You can't sell fire protection to Apartment 101 and deny it to 102. If you save 101 from burning, you're going to save 102.
So, a fire department like the one you describe would have a hard time selling its services due to the free rider problem.