Arguably, it never went extinct. Theres a Turkish guy, Dr. Miski, who thinks he rediscovered an extant patch of it that may have been grown in an old anatolian village. DOI 10.3390/plants10010102
Being frank, I didn't even realize this was available freely. When I searched "Miski silphium" the first place I ran across was asking for a log in (researchgate I believe) so I just copied the DOI since its easier for folks... PTSD from reading a lot of papers I guess. IYKYK
Anyways I first read about this in a book: The Lost Supper by Taras Grescoe. I also recommend checking that book out if you find this topic interesting.
Unfortunately, Silphium will likely never be resurrected. Even the Romans did not manage to grow it. Wiki:
Another theory is that when Roman provincial governors took over power from Greek colonists, they over-farmed silphium and rendered the soil unable to yield the type that was said to be of such medicinal value. Theophrastus wrote in Enquiry into Plants that the type of Ferula specifically referred to as "silphium" was odd in that it could not be cultivated.[16] He reports inconsistencies in the information he received about this, however.[17] This could suggest the plant is similarly sensitive to soil chemistry as huckleberries which, when grown from seed, are devoid of fruit.[2]
Similar to the soil theory, another theory holds that the plant was a hybrid, which often results in very desired traits in the first generation, but second-generation can yield very unpredictable outcomes. This could have resulted in plants without fruits, when planted from seeds, instead of asexually reproducing through their roots.[2]
PS: The poster (not me) seemed to have expected this. Throway...
The Romans may have had difficulty cultivating it but based on recent findings, local peasants in the region were capable of doing so. See my other comment above or check out DOI 10.3390/plants10010102 -- its possible it never went extinct at all.