What you didn't say is that the temperature attained by the surroundings as a result of a reaction would be independent of the amount of fuel. Because that would be false, right?
Your frozen wood ignites at the same temperature as warm wood, but heats the room less, because some of the heat that would otherwise raise the temperature of the surroundings warms the remaining unburned fuel instead.
In real fires, especially structure fires, heat accumulates over time, raising the temperature. The amount and type of the fuel is one of many important factors in how high the temperature of interior spaces can get. The "reaction temperature" as you call it is in no way a limit on how high the temperature of interior spaces can get. When oxygen is limited, fuel that's not burning yet gets pre-heated by the heat released by fuel that's already burning, and its combustion temperature is therefore higher when it eventually does burn. There is no inherent limit to this process.
Here's a very easy brain-teaser for you that illustrates the principle involved. You have a pump that can only produce enough pressure to raise water one storey. Is it possible to use this pump to raise water to the top of a 100-story building? (Hint: you have two buckets, and a very long extension cord.)
Respectfully,
Myriad