Assistance exercises are used to either correct or improve the execution of a lift.
Both are poorly understood and maybe it's our fault because it is also poorly defined and not uncontroversial.
Assuming that the movement skill is already automatic, according to the principle of specificity, a certain motor task (or sport movement) can only be improved by repetition of that task or a similar one.
We know it's not exactly like that. It was observed that certain other motor tasks improved the target movement. We call that transference.
The Olympic-Lifting-for-sports-preparation boom (it was always there but it became more visible in the past couple of decades) produced several success transfer stories.
Why does practicing the snatch improve volleyball or basketball players' vertical jump? Because it improves power output.
The other transfer that was known to physical therapy since the discipline's beginning is stability. And as I always say, "stability is the mother of strength".
Introducing some instability helps the athlete to more effectively activate the necessary muscle chains responsible for stability.
Unfortunately, that generated a wave of nonsense with infinite variations of lifting setups over unstable surfaces. The preferred arrangement involved a bright purple Swiss ball.
Again we have the example of an important principle that was completely misunderstood. While executing movements over unstable surfaces produces stability and coordination gains, being beneficial in some physical therapy and training cases, the stability and coordination gain is a result of higher core muscle activation. Maximal strength production is unaltered.
If we use a stable surface and unstable axial load, then we're talking: this type of motor task transfers to the target movement and improves maximal strength and the sticking point crossing time.