Before bringing in the final piece of the puzzle of spontaneity it would be a good idea to define this term we have been using so freely.
In ordinary conversations "spontaneous" is generally taken to mean something like "happens by itself". This certainly does not seem to describe the thermite reaction which requires significant intervention to initiate.
That last word of the previous sentence is important. All reactions require some minimum amount of energy to begin. Some require so little that we never notice (precipitations, for example). Some--like the thermite reaction--require a lot. But that energy is not part of the spontaneity puzzle.
In simple terms a process is spontaneous if it proceeds on its own once initiated.
In more technical terms, a process is spontaneous if it transfers free energy from the system to the surroundings.