I would not make the barrel-top temperature a design goal... you could push the system to the point where it may warp or damage the barrel without seeing significant improvements in more important factors like fuel efficiency or comfort.
It's useful to target an optimal, quite-hot temperature in the firebox (where it affects clean combustion) - for wood this is between 1000 and 2000 F, I understand coal burns hotter but if you get much above 2400 F you are burning nitrogen from the air and it starts creating a new kind of pollution.
The other goal is to store as much heat as possible in the bench. To that end, you want to move exhaust smoothly from the barrel into the heat-exchange mass without wasting too much heat along the way. But some heat loss in the barrel is useful to induce the exhaust to draft downward. It's all a balancing act.
The only reason I can see wanting an extremely hot barrel top would be for certain kinds of cooking.
There's also been some success with cooking over the firebox feed
itself, when using coal, if you can maintain good air flow to keep the fire happy at the same time. See this thread: