Charcoal - which you get from campfires is hot enough. It takes a lot more of it though and a lot of other effort. when bronze is available it is generally good enough and a lot easier, but historians tell me iron was used throughout the bronze age in small amounts. iron really needs steel to be signicantly better than iron and that took a while-
(I am a layman here so take this with a grain of salt) I believe you are correct, however no campfire will have enough airflow to get that hot unless you have a bellows or some other way of injecting air into it, and you'd have to have it structured in a way that it can efficiently burn the fuel. I'm not much of a blacksmith but had a friend who was into it and whose dad also was, and we did a pretty fair amount of "experimenting" as kids :-D I know from experience that elevation makes a big difference too, though I've never measured.
Would be fascinating for someone with knowledge of this to weigh in!
Charcoal is great for forging, though as you say getting airflow is tricky. Still this is manageable - clay and rocks are abundant on earth so there are options.