Social stability is not just about avoiding mobs, of course - it's also about keeping crime down, ensuring that pressure for change happens through peaceful channels, etc. But the sight of mobs rampaging through the streets is perhaps the most visible failure of states to maintain public order, and is surely a significant piece of what we ordinarily mean and care about under the banner of "stability".
(This is particularly true for theorists at the more cynical end of constructivism, who may regard society as an implicit compact between the proletariat and bourgeousie, in which the proletariat are granted certain rights and privileges in order to stave off revolution.)
I intend this not as a criticism of egalitarians, but as a debating point within egalitarianism. There are a wide range of factors which may plausibly reduce or exacerbate the risk of social disorder, and which have implications for the kind of egalitarianism one would endorse:
- The perception that the masses have little to lose from rioting (suggesting a need for broad-based prosperity)
- The perception that social elite possess large quantities of goods worth taking (suggesting a need for levelling down)
- The perception that social elites acquired their wealth unfairly (suggesting a need for equality before the law, and possibly more besides)
- The presence of intelligent and hard-working people who are unable to succeed within established institutions (suggesting a need for equality of opportunity, but not for equality of income)
- The perception that public authorities are biased against particular groups within society (which could point in a number of directions)
...and so on. Moreover, it matters how these interact. For example, one very simple (and highly dubious) model might suggest that all five of the above factors are individually necessary for riots to start. (An uncontroversial case of this is fire, where fuel, energy, and oxygen are all individually necessary for a fire to start.) In this case, we could focus on abolishing only one of the causes - perhaps whichever was cheapest to treat, or whichever cause we find most distasteful for other reasons.
A more realistic model would seek to quantify inequality of various types, and would give them different weights for their contribution to the frequency of riots starting and to the damage caused by riots. It seems likely, then, that if the achievement of social stability requires state action this is likely to require action against multiple different notions of inequality - we cannot focus on a single form of inequality.
I think this is far more a critique of egalitarian theory rather than of practice. John Rawls may have attempted to reduce equality to "sufficient and equal civil and political liberties, and maximisation of the primary social goods available to the worst-off class within society", but actually-existing democratic states are concerned to reduce economic equality, racial inequality, inequality of access to political institutions, ensure equal treatment before the law...
Speculatively, we might also take this an argument against the idea that inequality is intrinsically bad. Theories of the importance-in-itself of equality tend to end up focussing upon one particular conception of inequality. I think we have here a strong argument that no one conception of inequality is likely to satisfy all of our intuitions about the importance of equality, and ultimately the best evidence for egalitarianism - as with all moral theories - is derived from our intuitions.