In my experience your pretty accurate when you say the cooler moderates temperature changes but doesn't do much to keep things cooler. Makes sense. Insulation just slows heat transfer.

You can further slow temperature fluctuation by adding insulation and thermal mass. I like a kind of reflectorized, like space blanket, bubble wrap to add insulation. I make pouches out of it and then place the pouch in a cooler.

The easiest way to add thermal mass is to stuff a couple of bottles inside the insulation. Most kits benefit from a few liters of water. You can limit the top end somewhat by finding cooler places to keep the kit. In my truck the coolest spot is on the floorboard. If you use the floorboard find out where the catalytic converter is and avoid placing your kit near that spot. Catalytic converters can get very hot.