what prevents it from having more kinetic energy then when it started? What prevents the charges from being accelerated?

Let’s suppose that the KE of the charge carriers does increase around the circuit. That implies that the velocity increases. Since the current is proportional to the velocity of the charge carriers, that implies that the current increases. Since the current increases, by Ohms law the energy lost in the resistor increases. This reduces the amount of energy available for KE to increase. This process continues until a steady state is reached where all of the energy is lost in the resistance.

So what you suggest in fact is a transient state which leads to the steady state analyzed by the text. It happens every time you close a switch.

what prevents it from having more kinetic energy then when it started? What prevents the charges from being accelerated?

Let’s suppose that the KE of the charge carriers does increase around the circuit. That implies that the velocity increases. Since the current is proportional to the velocity of the charge carriers, that implies that the current increases. Since the current increases, by Ohms law the energy lost in the resistor increases. This uses up more of the PE reduces the amount of excess energy available for KE to increase further. This process continues until a steady state is reached where all of the energy is lost in the resistance.

So what you suggest in fact is a transient state which leads to the steady state analyzed by the text. It happens every time you close a switch.