Answer
For most polyprotic acids, the $K_a$ for the first ionization is much larger than the same for the second ionization or even a third one if it is possible.
Therefore, the amount of $H_3O^+$ produced by the second and third ionization is much smaller than the same produced by the first one.
In addition to that, the formation of $H_3O^+$ by the first ionization shifts the equilibrium to the left (Le Chatelier's principle), reducing even more the other ionizations, and making them even more negligible.
Work Step by Step
For most polyprotic acids, the $K_a$ for the first ionization is much larger than the same for the second ionization or even a third one if it is possible.
Therefore, the amount of $H_3O^+$ produced by the second and third ionization is much smaller than the same produced by the first one.
In addition to that, the formation of $H_3O^+$ by the first ionization shifts the equilibrium to the left (Le Chatelier's principle), reducing even more the other ionizations, and making them even more negligible.
