It arguably did lead to the extinction of the Sith (at least in canon), since both the Master and apprentice were killed at the end of Episode VI (Sidious and Vader, on the Death Star). Note that Kylo Ren is not a Sith; the Visual Dictionary for The Force Awakens states that Kylo
is the archetype of a new generation of dark side users that have emerged to fill the void left by the Sith's demise.
The danger of extinction was contemplated by both Bane (the creator of the Rule of Two) and his apprentice, Darth Zannah. The most relevant quote we have on the matter comes from Zannah, as she considered the possibility that Bane would die before her training was complete:
If [Bane] died now, everything he had put into place over the last ten years died with him. Zannah would have to start at the beginning. She would have to find and train an apprentice, even as she was still learning the full extent of her own abilities. She would be stumbling blindly forward, beset by enemies on all sides. It was almost impossible to imagine she wouldn’t make a mistake that would lead to her downfall … and the extinction of the Sith.
She couldn’t allow that to happen. For the sake of their order, she had to keep him alive.
Rule of Two, p. 291
Bane also considered the possibility of the extinction of the Sith indirectly while worrying about the implications of his failing body:
An apprentice had to earn the title of Dark Lord, wresting it from the Master in a confrontation that pushed them both to the edge of their abilities. If Zannah intended to challenge him only after he was crippled by illness and infirmity, then she was unfit to be his heir. Yet Bane was not willing to initiate their confrontation himself. If he fell, the Sith would be ruled by a Master who did not accept or understand the key principle upon which the new Order had been founded. If he was victorious, he would be left without an apprentice, and his failing body would give out long before he could find and properly train another.
Dynasty of Evil, p. 11
If Zannah acquired the title of Dark Lord by default (from simply waiting until Bane's body failed) she wouldn't have really earned it -- the Sith wouldn't be extinct, per se, but the survivor wouldn't have all the knowledge that a Sith should have (this is the same case that Zannah worried about if Bane died too early -- she'd probably fail and the Sith would become extinct). There was a similar problem if Bane defeated Zannah: his body would give out before he could properly train a new apprentice, again threatening the Sith with extinction.
However, the Rule of Two was no greater a threat of extinction than a large group of Sith, like the Brotherhood of Darkness. The problem with many Sith is that they would fight amongst each other and weaken themselves (this was, of course, the impetus for the Rule of Two). Indeed, the existence of many Sith doesn't really protect against the extinction of the Sith: at Bane's instigation, all the Sith except Bane were killed by their own thought bomb at the Seventh Battle of Ruusan!
Finally, it's worth pointing out that in some sense the Sith are an idea, and an idea can't be killed. The Sith leave behind holocrons which contain information about their beliefs and powers; as long as such holocrons survive then so will the Sith -- even if there are no living Sith. Sith holocrons have been responsible for the seduction of many Force-users -- even Jedi -- and could be used to re-create the Sith Order.