Looking at your diagram the main contacts are C with the overload on the motor supply, the 400 volt coil for this main contactor is C1 with one end connected to L1 . When you push the start button the other end of C1 will connect to L2 via the RESET contact, the N/C STOP button and the Brake control card on that Red wire via pins 3 & 2/4. This pulls in the main contactor C and auxilary contacts C2 & C3. C2 shorts out the Start button to maintain the circuit when the Start button is released. C3 is telling the brake control card that C1 is energised and the machine is working. When the STOP is operated C1 loses power and the main contactor C opens to stop the machine, at this point C3 opens and the brake control card loses power on pin 7 and knows the machine has been stopped so will now power the brake contactor coil BC1 via pin 1 which brings in the brake contactor BC . This connects the power card to 400 volts from L1 / L2 and injects a Dc voltage into the motor on L1 / L2 . With a three phase supply the motor has a rotating magnetic field that causes the motor to rotate, with a Dc input it creates a fixed magnetic field that will stop rotation and the amount of Dc injection will determine the rate of motor deceleration.
So it is possible to replace C with a new DOL starter that would be external to the existing enclosure and wire into the brake control card but you need to confirm that C is the root cause of your problem. Now if this was my machine this is how I would proceed.
1) Disconnect the grey wire from terminal 13 of the power card to disable the brake.
2) Is the main contactor C pulling in when you hold the START button in ?
If Not
3) Now with a decent DVM and test leads, set to measure Ac connect between A1 & A2 of C1 coil.
4) Operate the Start button and you should see 400 volts present which shows that C1 is powered.
5) ISOLATE the supply to the machine and measure the coil impedance between A1 & A2 of C1, if open circuit replace.
If C is pulling in but as suggested not all three contacts then again replace, but also are the auxilary contacts C2 & C3 pulling in ?