"Burn" and "melt" are completely different things.
"Burning" is a chemical reaction, usually with the oxygen in the air (or any oxidant, really). In organic burning, like the cake you suggested, the carbon compounds react with atmospheric oxygen, producing carbon dioxide, water (vapour) and (sometimes) an ashy residue.
"Melting", on the other hand, is a physical process in which the atoms of the thing you're melting can no longer carry more energy while remaining in the solid state.
The wikipedia articles for Burning ans Melting are pretty self explanatory, to be honest
Much of the vapor you see at that stage is unburned material, not a true "smoke" which would be ashes or non-burnable material. A true solid is very difficult to burn. Most fuels instead volatilize as the temperature rises, increasing the surface area. This material coming from the heated fuel appears similar to smoke.
A visible flame is the burning of this material. When the flame is present, there is less unburned fuel and less smoke. There are several videos where you can see the process of relighting a candle by lighting the "smoke" from an extinguished candle. (The material is candle wax, not smoke).
Wouldn't it become more efficient as it got hotter, and the level of smoke decrease as the fire gets hotter, until the flames appear?
Efficient burning depends on more than temperature. You can have a very hot interior, and all that will do is produce a lot of unburned fuel (visible as smoke) due to lack of oxygen. Inside the bulk, it is too difficult for oxygen to arrive rapidly to support efficient burning. But outside much more is available and the process can proceed to completion.
Right after the wood is introduced, it's cold and there's nothing to see. As it heats, it produces some volatiles, but perhaps not rapidly enough to actually burn. If it caught at that point, it might sputter out. As the temperature rises, the fuel begins to appear at a faster rate until it supports a full flame and combusts.
Best Answer
Burning faster could be understood in different ways. Here are a few comments:
Rate of chemical reaction
Being a chemical reaction, combustion rate usually obeys the Arrhenius equation: $$k=Ae^{-E_a/RT},$$ that is the reaction goes faster at higher temperatures, which help to break the reactant molecules.
Speed of flame propagation
Laminar flame speed gives the speed with which the flame front propagates along the mixture of the unburned reactants. It also increases with temperature: $$s\propto\sqrt{\frac{T_b-T_i}{T_i-T_u}},$$ where $T_b, T_u$ and $T_i$ are the temperatures of the burned and unberned reactants and the ignition temperature.
Adiabatic flame temperature
Adiabatic flame temperature characterizes the completeness of combustion process (i.e., how fully the reagents are used). It also increases with the reagents temperature: