> in general

That's too broad to answer. Let's narrow to our solar system, or a close approximation, first. Below, I'll step up to more complicated arrangements, and binary black holes. These are only three or four families of (regions of) general curved spacetime; there is a very large infinity of other families one could reasonably associate with "in general".

In a (model) solar system, there is a large central mass and several other objects with extreme mass ratios to the sun. There are interesting secular resonances among Mercury, Venus, and Jupiter, and between Earth and Mars, where that means that from time to time (over many orbits), the precessions of the near-to-the-sun parts of the planets' orbits match. These resonances may increase the eccentricity of the smaller mass's orbit, which can in extremis yeet the small body out of the solar system, or cause a collision between planets (e.g. Mercury's orbit may near aphelion cross Venus's orbit).

For the Mercury-[Venus-]Jupiter the closeness of Mercury to the sun tends to break up the resonance enough to significantly stabilize the shape of Mercury's orbit.

Any sort of slow-down on the inner body when it and heavier outer bodies are relatively close together with the sun is liable to suffice, and that's what the authors of the link at the top argue what we get with (good approximations of) GR. (The approximations used by the authors are a post-Newtonian expansion <https://en.wikipedia.org/wiki/Post-Newtonian_expansion> and numerical methods). Very roughly, thanks to the large central mass and the very different distances of the planets involved, the outer body rushes ahead or equivalently the inner body lags behind, which avoids stabilizing the resonance.

Many -- possibly even most -- stars are in binaries, rather than singletons like our sun. Planetary orbits in a binary (or triple) star system are likely to be dramatically different. In binaries, the width and mass-ratio between the partners are relevant quantities. Triples can be arranged in all sorts of ways. PSR J0337+17 is one known arrangement <https://www.youtube.com/watch?v=oDgfqq_W_uM>. PSR B1620-26, a pulsar-white dwarf binary, has at least one planet. I have only weak intuitions about what Jupiter or Earth mass bodies would do in such systems, or even how to select a reasonably astrophysical set of initial orbital values. (There are known circumbinary planets. See <https://en.wikipedia.org/wiki/Circumbinary_planet> and <https://sci-hub.ru/10.1007/978-90-481-8687-7> in which Chapter 9 details how these can be simulated).

It is not too fanciful however to suppose that periastrons could be close enough that post-Newtonian effects are non-negligible, even if the stellar masses are non-compact. That doesn't say much about whether secular resonances (in the sense above) are likely in such systems. I have no idea, but I'd guess they aren't forbidden.

> compare what % of systems are stable under Newtonian vs Einsteinian [gravitation]

That's what the authors do in their paper <https://arxiv.org/abs/2303.05567>, although "systems" here is a simplified model of the solar system (they discuss several simplifications and their justifications: the oblateness of the sun and the presence of moons are probably interesting "future work" but are most likely not going to result in large corrections see their §3.2).

For non-isolated systems, interactions with the environment can magnify whether these properties drive dynamical stabilization (merger) or instability (fly apart). The authors in their other fresh paper even more freshly summarized at <https://astrobites.org/2023/03/14/chaos-planets-future-of-th...> discuss how "environmental events" (nearby stars and their potential to approach our solar system) might lead to e.g. collisions between Earh and Mars, or an ejection of one or more planets into deep space.

Astrobites has also freshly summarized another "environment can change orbits" (for 2-body + dusty environment) at <https://astrobites.org/2023/03/13/will-it-merge-investigatin...>. ("Dust" in the GR sense of <https://en.wikipedia.org/wiki/Dust_solution>, rather than in the sense of fine particles of e.g. chalk that allegedly accumulate on sedentary GR theorists).

> Here's an experimental idea

Been done (and improving) for many decades. See for example Fig. 1 in <https://arxiv.org/abs/1105.1082v1>.