The reason why one often thinks that all the familiar phenomena can be explained just on the basis of the second and the first law of Newton is that it is clearly emphasized (mainly in school textbooks) that Newton's second law for a system of particles can take the form of $F_{external}=\dfrac{d}{dt}(p_{system})$ only when it takes in Newton's third law. Otherwise it would have been just $F_{net} = \dfrac{d}{dt}(p_{system})$.
The elementary form of Newton's second law is simply $F_{net}=\dfrac{d}{dt}(p_{particle})$. When one tries to derive the dynamic law for the total momentum of a system the second law helps to reach $\dfrac{d}{dt}(p_1+p_2+p_3+...+p_n) = \dfrac{dp_1}{dt}+\dfrac{dp_2}{dt}+\dfrac{dp_3}{dt}+...+\dfrac{dp_n}{dt}=F_{net1}+F_{net1}+F_{net1}+...+F_{netn} = {F_{net}} $. But the $F_{net}$ reduces to $F_{net external}$ only when one assumes that the forces of system of particles on each other cancelles out because they are equal in magnitude and opposite in direction. (Which is what we will call Newton's third law for the moment.) Actually, a stricter form of the third law asserts that these forces also lie along the same line of action and thus helps cancelling out terms of internal torque when one tries deriving the dynamic law for total angular momentum of a system.