(Under construction!!)
Q. One of the most common dismissals of range voting is that "first-past the post (aka plurality voting) lends itself to stability". This is difficult to counter; because it does lead to stable party structures, which in fact CRV's analysis shows, and countries with proportional or range voting systems generally haven't been around long enough to show decisive advantages.
A. This is a tough question which requires thought and research to answer... I.e: what is "stability"? How do we measure it? Is it desirable? And what would Range or Approval Voting do to it? None of these 4 subquestions seems to me to have any immediately clear answer... and it really is more than 4 since many more than 1 kind of stability could be considered...
One partial stab at an answer: If we have plurality elections, then via Duverger's law we get a 2-party system which is "stable" in the sense that other parties automatically die out. This causes a certain natural polarization.
If gerrymandering is allowed (as it is in the USA) and campaign contributions to those in office (even those without opponents) is allowed (as it also is in the USA) then the result can be 1-party domination where the other party is undemocratically gerrymandered and campaign-contributed out of existence. This is since campaign contributors love to contribute preferentially to those who already hold office – both since it works more like a bribe that way and also since incumbents have larger election chances in every government. Also since those in power get to do the gerrymandering. (Consequently in the USA we have 98% re-election rates.)
This also causes a certain "stability" – the stability of permanent unchallengable corruption! Hooray. Now when the power suddenly shifts to the other polarity we get a seismic thud to reverse direction. All the gerrymandering is suddenly redone, all the committee chairs evicted, and all the agency heads rubberstamp-approved the opposite way. This thud is the opposite of stability.
Compare that with range voting. Now we have more than 2 parties in power and a diversity of views. No one party can dominate and gerrymandering is less effective. A party can get effective dominance via forming a coalition, but that is a lesser form of dominance. When the dominance shifts, that is no longer seismic in scale – it is a coalition-shift and the third-party partners still may side the other way on some issues (similarly appointments are no longer rubberstamped and committee chairs need not be only of one kind) consequently the shift is not pole-to-pole but rather partial. That lack of an earthquake seems to me to be more "stability," not less, with range voting.
W.M. Herman & Michael Taylor: Party systems and government stability, Amer. Polit. Sci. Review 65 (1971) 28-37, showed by analysing a lot of real-world data, that governments with larger "effective numbers of political parties" tended to be "less stable" in the sense that they lasted for fewer days on average. However, this result is easy to misunderstand. Herman & Taylor when they said "a government lasted 160 days" did not actually mean that the government ended, e.g. was overthrown by a coup; they actually meant that the government continued on happily, merely under the control of a different coalition of political parties in parliament (and they only examined parliamentary democracies). Their work cast zero light on the real stability question, of what effect a multiplicity of political parties has on how long governments (actually) last. If there are more parties that can form more coalitions in more ways, of course the ruling coalition will change (on average) more often. This result was entirely unsurprising and thus uninformative. "Instability" in Herman and Taylor's sense is actually good since it means that a ruling party cannot maintain permanent control of everything, such as in Japan under decades-long corrupt single-party rule.