Is it correct to say that only binary and solitary star systems are stable? Recently it has been shown that a large number of binary stellar systems exist.  It seems to me that the orbits of stars in systems containing three or more stars will inevitably be chaotic, resulting in either mergers or ejections.  Binary stellar systems, though, are effectively stable (neglecting the very slow decay of orbits due to gravitational radiation and perhaps tidal interactions).  So, it seems that in the long run we should expect most stellar systems to be either binary or solitary; the first due to ejection (eventually) of all but two of the stars from a multi-stellar system, and the second due to the ejection of single stars from multi-stellar systems.  Is this correct?
No.
It isn't proven that any real orbital systems are stable on sufficiently long time scales.  Even our (single star) Solar System is not stable.  In their paper: On the Dynamical Stability of the Solar System, Batygin and Laughlin, use the numerical techniques pioneered by Laskar to show scenarios in which various planets fall into the Sun or are ejected from our Solar System on a timescale of 20Gyr.
The area of math that dominates long term orbital stability questions is Chaos Theory, which studies the types of systems of differential equations for which very tiny changes in the initial conditions cause very different resultant solutions.  Over long time scales, tidal forces, stellar radiation, and other energy sources cause orbits to evolve.  When resonances are reached, orbits can begin to de-stabilize.  Examples of this can be found in the Kirkwood Gaps of the asteroid belt.
Given the overwhelmingly high populations of binary star systems in the Milky Way (table below from wikipedia), we could probably call binary and single star systems stable at least at 10-100Gyr time scales.

However,Correa-Otto and Gil-Hutton point out that merely being in a galaxy causes binary orbits to be perturbed:

The main influence of the Galaxy on the WBS [Wide Binary Star systems] are
twofold: first, the disruption of the binary system, which produces
gaps in the orbital elements distributions of the three sets; and
second, the changes in the orbital configuration of the surviving WBS,
which modify the orbital element distributions of the population in
each set.

Given the paucity of discovered n-star systems where n is greater than 2, I think we can safely say these systems are mostly unstable at timelines on the order of 1-10Gyrs. In fact, many of them are unstable on much shorter timelines.