Technically speaking the crafting process is another item that is also subject to temperatures and while you’re crafting the food does get colder while you’re technically “cooking”. But in reality you’re just crafting, so temp rules apply. It’s also the reason why you can start with fresh ingredients, interrupt the process, leave it, and come back to finish crafting it and end up with a rotten result.
I think the ingredients start hot the moment you craft and get colder the longer the craft goes on. I recommend cooking in a heated room(little vehicle heater 1x1) in winter. It’s dumb, I agree, but it’s how it is.
As for not being able to use frozen items, I do not understand your issue as I use frozen meats, etc, all the time. Everything goes into fridges that freezes the edibles next to my cooking area and are always available for cooking, despite always being frozen.