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It depends on the K. When the products are more stable more will be produced, meaning that the K will be larger. When the reactants are more stable they will wish to not react as much, thus there will be more reactants and thus K will be smaller. So to answer your question, no, the product is not always more stable.
Reactions will try to reach the most stable state as it reaches equilibrium. If products are more stable, more products will be produced and its concentration will be higher than the reactant concentration. This leads to a Higher K value. If reactants are more stable, the reactants will not react as much and its concentration will be higher than the product concentration. This leads to a smaller K value.
Products are not always more stable than reactants. In his example the K value just happened to be large, meaning the reaction favors the products. Reactants can also be more stable than reactants, which would mean a smaller K value.