yes!!’
I let food cool, not because of ambient temperature, but because of condensation. I let it cool in a storage container, but with plastic wrap over it. While it’s cooling, it’s going to sweat, right? And I don’t want to put it away and let condensation build up, because that will seep into the food and affect its integrity/flavor. So when it’s reached room temp, I replace the now-damp plastic wrap with the dry lid, and then put it in the fridge.
The basic part of this is not all that complicated. Bacteria grows rapidly in food at temperatures between 40F and 140F. You don’t want to keep food in that range for more than 2 hours total before eating it. It’s worse at the higher end of the scale, so 1 hour between 100F and 140F is the max. If you put the food in the fridge which will be under 40F it will cool faster than at room temperature. So cool your food rapidly outside of the fridge in the ways described above, or put it in the fridge.
Those are the simple rules. It does get more complicated in the details. You may or may not have pasteurized your food when you cooked it. If it’s been hot enough for enough time it is effectively bacteria-free. It will take longer to spoil from that point at elevated temperatures. But you can’t easily tell how long, and if the food is exposed to the air it can acquire more bacteria that way. If you put a large hot item in your refrigerator it may spoil some small items near it if it is too hot, you have to do some balancing, and rapid cooling methods outside of the fridge should be used.
Just keep your food between 40F and 140F for as little time as possible.