A DC operated motor does not need a start capacitor or a run capacitor.
They do need new brushes eventually, and/or a cleaning of the commutator.
IF there are bad winding sections/bad commutator faces, you may have a motor that just tends to stop at a "bad spot" and then
it won't start as it is likely to always hang up at the point in rotation where there's no current flow.
Once above starting speed, it flies past the past the bad spot and keeps on going.
I'd put a meter looking at DC current directly in series with the motor.
See what the current is when starting vs running.
If it's tending to stop on a bad section, there will no or next-to-no current when calling for startup.
Does the motor EVER start properly?
It can be both worn bearings and a bad commutator sector or two.
This is commonly seen in the dc motors used in auto window motors.
When people hold the button down after the window has reached full travel, the motor draws heavy current once stopped, and this can either burn out a sector, or burn the face on the commutator. Sometimes this can be cleaned up
by burnishing the commutator, or replacing brushes.
A short term solution in a car window motor is slamming the door while holding the button in a run position.
This will often start the motor and it will run...but then they always seem to stop again at the "bad" spot.
It could also be high resistance in a relay that starts the motor. Bad contacts in the run relay will drop so much voltage that
current will never be high enough to actually start the motor.
If a sector is truly burned out, then the only answer is replacement or rewinding the motor.