I don't know if the CPS is the problem, could be.
It is a magnetic reluctance device, essentially an electromagnet ( wire wound magnet ) that is excited by the flywheel teeth.
It generates a sine wave signal at a frequency dependent on RPM. This signal is sent to the PCM via twisted pair wires.
( filing the tip would be detrimental, and if too much filing was done it could become inoperative )
It, and the Camshaft Position Sensor ( a Hall Effect sensor which generates a square wave signal ) are used by the PCM to determine the exact position of all engine components for ignition and fuel injector timing.

This is the reason that all modern engines utilizing this technology must turn over a minimum of 2 revolutions before actually starting.
( Takes 2 revolutions for the PCM to get it's ducks in a row )