In order to learn more about related phenomena, one needs multiple data points with varying initial conditions. In order to understand more details about a specific case, no further data points are needed or desirable. Graham's essay is about his interaction with, and opinions about philosophy; he doesn't need any other data points in order to tell you his opinion, nor do you need any other data points to talk about history.
2) Actually, one would nominally point to the start of both fields in Thales, who was seen by Aristotle to be the progenitor of Greek philosophy, and supposedly was the first person to note that the angle inscribed in a semicircle is always right. Note that, even before that, there definitely existed at least non-trivial arithmetical problems (such as finding the volumes of various solids), that were known literally a millennium prior (to learn more, consider the wiki pages for Thales, the history of mathematics, and Egyptian mathematics).
Of course, this is brushing under the carpet the fact that in Ancient Greece, the distinction between a "philosopher", a "mathematician", and a "scientist" did not yet exist (exception: physician was an independent job title); the Greeks could not have seen themselves as anything less than all three. Graham's claims have much more to do with the fact that philosophy, as an independent field, continued to exist long after this distinction became clear.
3) Your claim here is not entirely clear. Are you really trying to validate the failed methods of philosophy by saying that it's important that we try them now so that we don't make the mistake of trying them later?
4) Three of your examples date from 1750 to 1850, and are examples of how enlightenment thinking bridged the gap from philosophy into what could actually begin to be called science--these people were called philosophers because there was no other word yet extant to describe what they did, but we now recognize their results as belonging to other fields. The fact that they called themselves philosophers is irrelevant; their results did not in any way require the existence of the rest of the field of philosophy.
The important contributions of Russel and Whitehead to mathematics might have been inspired by their personal philosophies, but having such a philosophy does not require the existence, import, or study of the many misshapen works of the field.
Finally, Popper is an example of exactly what philosophy can do to actually be useful--answer a question about how to approach the gathering and use of knowledge.
In summary, what he wants philosophy to do is indeed science...so why are there still philosophers doing anything else?