This seems like an interesting thought experiment care to expand on that?
When an expanding menace tries to absorb the ancient stable culture, it has key disadvantages. Its attention is spread over activities on a large periphery, and its stores of knowledge and weaponry are limited. The ancient culture has been found by many menaces before, and survived them all. It knows of other old, stable societies, and they may even have evolved a common plan for dealing with menaces.
It is conceivable that the expanding menace would try to flow around the old stable societies, like a river around rocks, without threatening them enough to provoke a backlash, but that runs against the logic of expansion.
So, odds are any expanding menace will encounter one of them, or more than one, if needed, before it gets to us. Each such encounter will be an existential challenge at bad odds, which multiply out increasingly close to zero.
Thus, the stable condition of a galaxy is a sparsely occupied archipelago of deeply ancient, unassailable cultures not interested in expansion, so not visiting, e.g., us, with the occasional rash of menace, dealt with firmly.
As you adjust down you could trivially and likely have tens of light years between cultures or even hundreds on average if intelligent cultures are short lived or particularly rare. Expanding menaces could be automated and self replicating and designed to create an expanding zone of safety for an old stable society in a way that isn't easily traceable to the old stable society.
It's young because it's expanding. To first order, divide the radius by the rate of expansion and you get when it started. The society that bides its time for millions of years and then turns menace does not pass the smell test. Every year they go without menacing, the less likely they ever will.
Anyway, whoever's (1) inside the perimeter, (2) is a bad trading partner, and (3) not burnt is a pretty good bet for who's responsible. If there are two plausible candidates, they are better both dealt with.
It's also true that they don't last long because of conflict, but that's secondary.
A sorcerer's apprentice scenario is conceivable only in the short term. To maintain expansion demands adaptation to an increasing variety of new challenges, whose variety and strength increase as the fourth power of expansion radius. (Three dimensions for space, one for age of opponent.)
One can invent myriad obscure scenarios, of decreasing likelihood. It's a big universe, rashes happen, but it's a big universe, so they mostly happen somewhere else.
Stable societies that stick to their own region while prioritizing research ('growing tall' is the name for the strategy) tend to win out in battles against those that 'grow wide', grabbing all resources and expanding into them with little regard to research and personal affairs.
Those that 'grow wide' tend to win the games through simple attrition on a long time-span at the end-game phase. Simple resource volume tends to be able to win against vastly superior forces, much like a StarCraft 'zerg-rush', and the superior force usually loses in a 'death by a thousand cuts' fashion -- but individual battles before this phase of the end-game are almost exclusively won by small civilizations that 'grow tall', much like the 'stable culture' of your example.
The 'Growing Tall' strategy tends to be the one chosen by players who like to end games early and play aggressively against the other players -- contrary to the sedentary and passive method of play that the strategy facilitates.
It seems like being equipped to, say, cause a solar flare to sterilize an offending planet would tend to discourage an aggressor before their ships got used up. Probably a flare wiping out a neighboring planet (e.g. one like Venus) first would convey the message adequately.