Hi,
What you are missing is very simple. You seem to be assuming that torque is ONLY required to overcome INERTIA. Torque is ALSO required to overcome FRICTION.
In other words, what you are explaining yourself is only that which happens when only the inertia is considered in a system and the friction is ignored.
What you are considering is the torque that has to overcome the inertia only. To get acceleration the motor has to push harder because the inertia acts against change.
But after the motor comes up to speed and the speed remains constant, there is still friction in the system that requires constant push from the motor.
The friction and the inertia enter into the equation as two different constants. If you decrease the friction to zero then you dont need torque to keep the motor running, but that's not considered possible in the real world.
You can think of it as a brick flying through space vs sliding across a glass surface. The brick flying through space requires acceleration to get to speed, but after that no more force is required. The brick sliding on the surface however even after it comes up to speed needs constant force to keep it going because it is rubbing on the glass surface (even though the glass is smooth there is still some friction present).
So the bottom line is that there are at least two things that require torque from the motor:
1. Inertia
2. Friction
The inertia is something that requires an application of torque for a limited amount of time assuming the motor reaches some constant speed, but the friction is something that requires constant torque even after reaching the final speed.
Another way to think about it is like this...
Say you have a 'perfect' motor with no friction, but still has some rotational inertia. You start the motor up, it comes up to speed slowly, then reaches the final target speed, lets say 100 rotations per second. With no friction, the rotational inertia will keep the motor spinning indefinitely. This mean you can drop the motor current to zero (and thus have zero torque) and still have the motor keep turning.
But now you lightly touch a finger to the shaft of the motor, which causes friction between your finger and the shaft. What happens? Obviously the motor experiences friction now, which would tend to slow it down unless more current was delivered to the motor to keep it running at 100 rps as before, and this is because the current develops more torque in the motor. Thus the motor again runs as 100rps, but only with the addition of MORE torque.
Also, remove that extra current and that takes away that additional torque, which would then cause the motor to slow down, so it becomes clear that more torque is needed to overcome the friction. Since the friction is constant it requires constant torque to keep the motor up to speed.
A simplified model of the motor with no inertia and no inductance:
w=Ia*R*Km/(F*R+K)
where
w is the speed,
Ia is the armature current,
R is the armature resistance,
Km and K are motor constants,
F is the friction.
Since F is in the denominator, as F is increased it takes more and more current (Ia) to keep the motor at the same speed (w).