About spontaneous combustion of methane
&quot;Spontaneous&quot; means different things in different contexts
Your penultimate paragraph captures a key idea. The explanation for why this is right requires a recognition of the context of the term &quot;spontaneous&quot;.
The context of the statement at the start of the question $\Delta G=\Delta H-T\Delta S$ is negative is thermodynamic stability. But this is somewhat at variance with the more natural use of the term which implies &quot;things happen without being pushed&quot;. This idea is closer to the idea of kinetic stability in chemistry.
You correctly identify the need to add energy to get the reaction past the transition state. Even if the reaction overall releases energy (thermodynamically spontaneous) the reaction won't just happen if there is a huge barrier to getting over the transition state. There are big kinetic barriers that stop the reaction &quot;just happening&quot;. Oxygen and gasoline will react to release energy but this doesn't happen without the push given by the spark plug in the engine of a car.
That barrier is so low in some cases a compound will react with nearly everything with little excuse (chlorine trifluoride will set fire to asbestos). That is a spontaneous reaction in any context. Luckily, few thermodynamically spontaneous reactions are also kinetically spontaneous or humans would catch fire in air.
So when you see the term &quot;spontaneous&quot; ask what is the context: thermodynamic or kinetic? And don't confuse them.