The Romans had a few different contraceptions. Soranus believed that "to smear the entrance to the uterus with olive oil or honey or sap from a cedar or balsam tree, alone or mixed with white lead" would work. Pliny the Elder in his Natural History states, "There is a third type of spider, called a hairy spider, which has a very large head.If this is cut open, one finds inside, it is said, two small worms. If these are tied on to women with a strip of Deerhide, they will no conceive.

Source: Jo-Ann Shelton, As the Romans Did: A Sourcebook in Roman Social History. New York, Oxford university Press, 1998.

I am specifically reference the primary sources in chapter 2, which are Pliny the Elder's Natural History and Gynecology by Soranus.

Hope this helps!

Did the Romans really drive a plant species to extinction based on its contraceptive qualities? Are there any primary sources that corroborate this? Is it possible to know if the plant was effective?

You're thinking of Silphium, which is a somewhat mythical plant in that we aren't sure what plant it was so we can't say for certain whether it was harvested to extinction or even worked.
Though it is suspected that Silphium is/was part of the genus Ferula and other plants in there have been found to have medical effects. A study on Ferula asafoetida found an 80% reduction of pregnancy in rats with oral administration.

We have several contemporary writers describing the effective usage for the plant. Pliny the Elder's Naturalis Historia for example essentially has it's own chapter on it, which reads more like a spam mail with all the ailments this plant cures. But it was by far not the only method, Soranus of Ephesus writes in his Gynaecology about several supposedly effective recipes to avoid or terminate pregnancies with, most of them make no mention of Silphium (which he calls Cyrenaic juice, after the Libyan region it's from). Example:

Of unripe oak galls, of the inside of pomegranate peel, of ginger, of each 2 drachmas, mold it with wine to size of bitter vetch [peas] and dried in the shade and give applied as a vaginal suppository after intercourse.

Dioscorides' De Materia Medica also has a lot of recipes depending on what you want to do. Silphium and pepper as oral contraceptive and a variety of plants if you want to abort a fetus (myrrh with wormwood, lupin and rue; birthwort with pepper and myrrh; great arum; etc.)

For late-term manual abortions we can go back to Soranus and his Gynaecology. With a big, bold NSFW as it mainly involves chopping and crushing the embryo in the uterus:

[...] one should split it with an embryotome or a knife for removing polypi, covered during its introduction between the forefinger and the little finger, so that the fluid may be emptied and the circumference of the head collapse. If, however, the fetus naturally has a large head, one should crush the head with the hand, <for> it yields easily when the body is still soft.

Quote: Gynaecology, book IV, page 193 (Google books doesn't display p.192 so the first sentence is cut off) - you can read on here if you really want.

Sources:
Contraception and Abortion from the Ancient World to the Renaissance by John M. Riddle
Gynaecology by Soranus of Ephesus