Good question!
In the Home Operations Discord there's some very smart people who solved this problem inside kubernetes by checking if their NAS is online (through a Prometheus exporter named node exporter) and then scaling down their workloads that use it, automatically, using KEDA (an autoscaler for kubernetes)
Depending on how your processes are orchestrated, you might be able to do something similar?
Random uneducated guess: Could it be some cache/shaders stuff? Depending on the game, it could take a while until the cache is warm and everything runs smoothly.