The big capacitor limits the duration of the pulse that is sent to the clock. The function of limiting the duration of the pulse is what is meant by "pulse-forming".
It would also be possible to arrange a circuit that only allowed the output transistor to turn on for a fraction of a second, and then the output transistor could be directly connected to the clock input. That is what is meant by moving the pulse forming in front of the output transistor.
On a related subject, the existing design puts out a positive and a negative pulse. Is that negative pulse is needed or is it just to discharge the capacitor that becomes charged during the positive pulse? If the negative pulse is needed, does it have to be negative, or could it be a second positive pulse? If there is a circuit that has the pulse forming before the output transistors, knowing the answers to those questions could allow the circuit to be a lot simpler.