This conclusion does not seem to be the simplest conclusion from the evidence. Forest fires would be more rare in cold periods, but they would still occur - if they happened 10 times less often you'd expect to see around 10% as much fire usage, not 0%. While it is possible Neanderthals in the region abruptly stopped using fire in the region for tens of thousands of years, another possibility is that they started making a different style of hearth which did not leave the same form of evidence. Perhaps they switched to making fires in different parts of or outside the cave, or perhaps they built fires on raised stones that provided better airflow but also meant buried artifacts recieved less heat, or perhaps as sediment built up in the cave they started digging pits for their hearths which would make them appear older, etc.
While none of this explicitly rules out the possibility that Neanderthals lacked the ability to make fire, that a hominin so similar to us that we could potentially crossbreed could survive in arctic climates without fire is an extraordinary claim, and absence of evidence for fire at these locations is not necessarily evidence of absence.
While none of this explicitly rules out the possibility that Neanderthals lacked the ability to make fire, that a hominin so similar to us that we could potentially crossbreed could survive in arctic climates without fire is an extraordinary claim, and absence of evidence for fire at these locations is not necessarily evidence of absence.