Would the Rule of Two eventually make the Sith vulnerable to extinction?

Question

When a Sith apprentice defeats the Sith Master, there would be for a while just one Sith Lord. While there is only one Sith, the Order is vulnerable. That one Sith could later die of wounds sustained in the duel with the previous Master. The remaining Sith could die of other causes. Or the remaining Sith could decide the whole dark side lifestyle was just too risky.

Or consider that at least one Sith Master was killed by somebody other than the Apprentice when Darth Caedus was killed, and his apprentice turned away from the Dark Side.

The Sith may have endured a thousand years with the Rule of Two, but eventually their luck would run out.

The Rule of Two is supposed to guarantee that only the strongest Sith survives, but that's no guarantee the Sith Order survives. It seems to guarantee the opposite.

Answer 1

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.

Answer 2

The rule of 2 does not prevent other darkside force users, it just says they can't take on the mantle of a Sith Lord while another is in office and they take their orders from said Sith Lord.

The idea of the rule of 2 was before that it was common for the apprentices to end up all fighting the master to get rid of him and then the strongest ones would fight to become the new master. With the rule of 2 it consolidates power in the master and helps him ensure the only one he has to worry about being overthrown by is a single apprentice.

The interesting thing is that Snoake and Kylo have not taken on the moniker of Sith lords themselves with the name Darth, there is nothing stopping them as top of the pyramid in The Force Awakens. There are possible reasons for this 1) there are other factions vying for the position that are not in the movie, 2) They do not yet consider themselves worthy (this is unlikely from typically egotistical dark side force users), 3) They are going back to what Darth Sidious and his predecessors did during the republic and not using the title publicly to avoid being a target, while they know the Jedi order is currently still in shambles they can possibly see it making a recovery in the near future, however their actions aren't helping them in this point either.