Is the motor inductance enough for LPF action?
Your model of the motor is wrong. There should be also a voltage source which presents the induced voltage which in simplest models is considered to be DC which is proportional to RPM. It's not clean DC, it's approximately sine tops which are separated by peaks and oscillations caused by inductance and commutation.
Consider to use this model as a simple approximation:
The thick vertical peaks in the induced voltage Ui curve present noise and oscillations caused by commutation and inductance, actually by inductance La, but no harm, it's summed to the by rotation induced voltage. Nothing guarantees that there's sure clean zone between the noise bursts. If the brushes are worn noise can be continuous. The average of Ui is coarsely proportional with the rotation speed (=RPM).
The noise and systematic waving which exist in Ui can be seen also in motor current. The noise is probably a little reduced because in La induced peaks occur as voltage which brake the changes in the current. But long enough poor contacts surely can be seen.
You need quite effective lowpass filtering to get usable motor current estimate. How much filtering- it must be tested. I guess your PWM frequency is so high that its filtering needs are cured without extra effort. But that's a guess only.
Note: You need an estimate of noiseless Ui to know the rotation speed. It can be calculated in the program or it can be formed with analog circuit which gets the motor input voltage and the current.
Start by measuring the parameters of the motor. To get the proportionality between RPM and Ui you must rotate the motor with known speed. It's also possible to measure reached speeds with different input voltages.
Friction torque makes the motor to eat some current without any external load (that torque is unavailable for external loads). Friction depends strongly on RPM in a complex way, so to get the induction modelled properly without tricky math rotate the motor and use it as a generator.