Then, typically, fires exist in a state of increasing starvation? If the oxygen around a fire were to be deliberately added at a steady rate, the fire would continue to flare?
The rate at which a combustion reaction proceeds may be limited by the concentration of oxygen, the concentration of fuel, or the amount of available heat. When the speed of a reaction is limited by the concentration of oxygen or fuel, it can be controlled by regulating the rate at which oxygen or fuel are made available, since the faster the reaction occurs, the less oxygen or fuel will be available to it. When the speed of the reaction is limited by the amount of available heat, however, it is very unstable. The faster the reaction occurs, the more heat will be available to it; this process will often accelerate until the rate of combustion is limited either by the availability of oxygen or fuel.
Many materials are not directly flammable, but when heated will decompose into other materials that are. Often, the decomposition reactions may occur at temperatures below the ignition point of the substances produced, and may absorb a considerable amount of heat. As a consequence of this, it is entirely possible for a fire to give off substances which decompose into flammable gasses or particulates, but which lose so much heat doing so that they can accumulate without igniting. This accumulation may continue until some of the materials are hot enough that--even after decomposition--they are hot enough to auto-ignite. This will quickly heat up nearby materials which were almost--but not quite--hot enough to auto-ignite. This process will proceed very quickly until it consumes nearly all the immediately-available oxygen or fuel, whichever runs out first. Once that occurs, the rate of the reaction will be limited to the rate at which oxygen or fuel is made available to it.
More oxygen accelerates the rate of combustion, so yes, I think you’re correct.