You’ve hit upon an excellent point. When an exothermic reaction happens, the energy starts off in the system (with the products moving or spinning at high speed). So the system is extremely hot immediately after the reaction.
I’m defining the system as ‘those entities involved in the reaction, i.e., the atoms/ions in the reactants and the products’.
The system is anything else.
This excess thermal energy is transferred into the surroundings gradually, through conduction/radiation. Overall, the system loses thermal energy, so the reaction is exothermic. The system does not cool below the temperature of the surroundings.
Just because a reaction is described as exothermic does not mean that the transfer of energy is instantaneous.