There are murmurings of a possible legitimacy crisis if a highly partisan and/or constitutionally intemperate individual like Kavanaugh gets into the Supreme Court.
However, the Supreme Court has been highly suspect of compromise for some time, and carries with it a dubious legacy of a large number of questionable decisions. The problem is that it takes some knowledge and argument to cut through the ideological fog and political propaganda that papers over decisions like Citizen United and Bush v. Gore and a plethora of others.
Knowledge and argument take energy, which is limited to a given proportion of capable witnesses, so issues of legitimacy that require more of this type of thing are far easier to conceal.
Someone like Clarance Thomas is just as compromised as Kavanaugh, and has just as much of a troubled past, as Anita Hill made clear. But he also has more self-control over his own external persona, and he keeps his mouth shut, so he passes under notice of most, and the establishment media allows this as well.
But for someone like Kavanaugh, there is an open and visible defiance of those traditional mores that govern an institution like a judge's seat. This is more difficult to hide. And this is coupled with an ever-growing interest in political affairs in the body politic, as the large viewership of the hearings - which are protracted and procedural, and which have been ignored by many in the past few decades - has made clear. And humans being what they are tend to take the visible and perceivable behaviors and procedures of the powerful as signals for conducting their regular affairs, at least as far as they aren't threatened by double standard laws and enforcement.
So a legitimacy crisis does grow as it becomes more visible to a greater mass. And this crisis is further intensified by lost trust, intensifying public interest in their own immiseration, and a long history of delegitimation and escalation on the part of the political right as well, which all add to a volatility which feeds the crises themselves.
