My real question is in fact "How can dissolution happen if it's endothermic?" However, there have been many questions regarding this exact topic and the answers about Gibbs free energy determining spontaneity of the reaction doesn't help my intuition at all. The majority of answers list the increase of entropy as the cause of endothermic dissolution, but in my opinion, entropy doesn't make sense as the cause.
Let's take a similar problem, namely the melting of a solid. If we place a solid with lower temperature into surroundings with higher temperature (enough for the solid to reach its melting point), it's not that liquification would cause the temperature of the surroundings to drop, but rather that the transfer of heat from the surroundings to the solid would cause liquification. The acquired heat allows the solid to increase its entropy, not the other way around.
So, if increased entropy is the result of endothermic dissolution, what is the cause? Or, what is the mechanism that allows entropy to be increased?
Let's take the dissolution of NaCl in water as an example. This process is endothermic by around 4 kJ/mol. My guess is that, since some ions in the solid have higher kinetic energy than average, this energy will be sufficient to overcome the 4 kJ/mol barrier and leave the solid, taking some of its thermal energy away from it. As a result, the temperature of the remaining solid will drop a little. Since the temperature of the solid is now slightly lower than the temperature of the surrounding water, heat will be transferred from the water to the solid until the temperatures are equal. This will result in lowering the temperature of water and in the increase of entropy. Is my thinking correct? If not, why?