Yes, generally, the things that kill an unpowered probe are the physical changes associated with extremely low temperatures. Even if the processor itself continued working correctly, at the extremes of cold we're talking about, the differences in contraction between materials can pull apart components. I'm talking about things like the metallic circuit traces breaking off of a circuit board, wires snapping, possibly even solar panels delaminating or battery cases cracking open.
In addition, extreme cold and sitting in a deeply discharged state for a long time can permanently damage the internals of batteries, making them impossible to use later even if charging voltage becomes available. For example, lithium-ion batteries are famous for being difficult to recharge if they drain deeply enough, and any liquid electrolyte solutions are subject to freezing, which could both physically damage the battery and change the qualities of the solution when it starts to defrost under sunlight later. It may not be possible to design a battery that would still produce power at the lunar night-time temperature of -200 °F (-130 °C) without heaters, regardless of what conditions the processor is capable of functioning under.
You mentioned avoiding heat/cool cycles with insulation. But, there's not really any way to go from full sunlight to deep-space shadow without heat/cool cycles. There is no insulation in the world that could maintain temperatures across weeks of constant direct multi-hundred-degree sunlight and weeks of total darkness. Just look at how much work the JWST has to go through to keep its sensors at deep-space temperature, and that's with the entire design revolving around pulling off that trick, with constant power to run the coolers. Maintaining that sort of cold while still having attached components sitting in constant sunlight is a non-trivial problem, and that without actually sitting on a surface that's being baked by the same light.
To avoid getting damaged by freezing, most solar-powered probes use heater circuits to stay warm when they're going to be in a shadow for an extended time. But heaters cost battery power, and that puts a hard limit on how long they can be shaded before they have to shut down and risk taking damage from the cold. Most probes will have a 'safe mode' where they can kill the heaters and just wait for dawn with minimal power usage (it's actually the hibernation mode they were using during the flight out, in most cases), but it's generally expected that something will go wrong and it won't ever wake up. And sometimes the probe needs to actively keep an antenna pointed at Earth to keep talking to us, so even if it does wake up in the morning, it may not be able to reacquire and get stuck yelling into the void. So most lunar landers just assume that sundown means the show is over. (Of course we still hope. Chandrayaan 3 is currently shut down waiting for dawn. We hope it wakes up. We don't count on it.)
Now, the alternative is using radioactive materials for heat and/or power. A few lunar rovers, including the USSR's Lunokhod 1 & 2 and China's Chang'e 4, have used radio-thermal heaters to survive the night and continue their work in the morning, but in all those cases the radioactive decay provided only heat, with electrical power depending entirely on solar panels. (The downside there is the cost and the weight, not to mention the risk; the first attempt at a Lunokhod lander blew up on launch and scattered radioactive polonium across a wide area.) An RTG — a radioisotope thermoelectric generator — uses the heat of decaying radioactive materials to produce electricity. RTGs have been used on most of the probes heading to Jupiter and beyond, where the weak sunlight is not a viable source of power, and on a couple of Mars rovers (as made famous by The Martian). In that case, cold is no worry at all; usually in space, the problem is getting the RTG to radiate away enough energy that it won't melt!
But RTGs and similar technologies are usually not used on the moon. The costs and risks would have to be justified, and the moon just doesn't seem to do that. The moon is a long way off, but it's way, way closer than Mars or Jupiter. Astronomically speaking, it's easy to get there. It's probably more beneficial to land a probe on the moon, gather data for two weeks, and then shut it down and use what we learned to build a new probe that can do new experiments instead of repeating the same ones for months or years on end, even if they were fairly interesting experiments.