Take the example of formation of water from Hydrogen, Oxygen this reaction highly exothermic but kinetically slow
The reaction to proceed first the reactants have to cross their activation energy barrier so that they form products giving the energy. Hence catalyst or temperature of the process plays a role in reaction to proceed spontaneously
The other example is burning of LPG gas requires ignition for reaction to gets started even though reaction is spontaneous.
Here is a video that explains spontaneity and signs of enthalpy and entropy: