don't the neurons already have plenty of serotonin stored in the axon vesicles from which the neurotransmission happens
Sure, but neurotransmission is happening all the time. It's not uncommon for a neuron to fire several times a second, and though not nearly every vesicle is released per action potential, over minutes to hours you should be thinking about hundreds to tens of thousands of possible release events.
Of course, neurotransmitters aren't simply released and then gone; monoamine transporters are particularly famous drug targets; these transporters facilitate reuse of neurotransmitters as well as their degradation.
I had a bit of trouble finding a solid estimate of brain half-life/turnover of serotonin in papers I have access to, though this paper used an indirect method by using an inhibitor of subsequent serotonin metabolism:
Udenfriend, S., & Weissbach, H. (1958). Turnover of 5-hydroxytryptamine (serotonin) in tissues. Proceedings of the Society for Experimental Biology and Medicine, 97(4), 748-751.
Such estimates indicate a half-life in brain of the order of minutes
Other papers like this one:
Diaz, P. M., Ngai, S. H., & Costa, E. (1968). Factors modulating brain serotonin turnover. In Advances in pharmacology (Vol. 6, pp. 75-92). Academic Press.
refer to
Reports in the literature show that the turnover time of brain NE stores is 5.8 hours and that of brain 5-HT stores is 1.2 hours
but I don't have access to the full-text to find what reference they refer to. However this seems consistent with the timeline of tryptophan depletion (over several hours, or days) referred to in papers like this one:
Nutt, D. J., Forshall, S., Bell, C., Rich, A., Sandford, J., Nash, J., & Argyropoulos, S. (1999). Mechanisms of action of selective serotonin reuptake inhibitors in the treatment of psychiatric disorders. European neuropsychopharmacology, 9, S81-S86.