An interesting question rakuz66, and one which needs an answer. I will take a stab at it. The chimney flue is the interior of the liner. It is running inside another pipe, or set of tiles stacked and mortared, or bricks and mortar or wattle and daub. The cap sits on top of the chimney chase, which is the aforementioned tiles, etc.
In operation the interior of the flue gets hot, and it gets hot. The surrounding air gets hot, and you want that to happen because you do not want the liner to get too cool as the gasses get cool, and the components in the smoke condense on the walls of the flue. You want a warm flue, all the way to the top.
There is a secondary air flow, between the liner and the chase. This flow needs to be minimized because the flow is generally going to be from your living space to the underside of your chimney cap which is unlikely to be hermetically sealed. Air flowing outside the flue will cool the flue. Most people attempt to minimize the amount of air flow lost up the chase through the space between the liner and the chase. This is most easily done by packing properly rated
insulation around the top and the bottom of the liner as it enters and exits the chase. The air surrounding the flue will heat up, and dead air is insulation without a bunch of non flammable fibers in it. In such an installation there is no air flow to speak of, and even those who worry about that little bit could not subsist on it. It is certainly less than that lost through your attic insulation due to the small area and the density of the packed insulation.
Other folks hermetically seal the cap in place, and fabricate sheet metal plates to plug the bottom of the chase and then they insulate those. This keeps hot air from filling the smoke chamber of an old fireplace and escaping up the chimney between the liner and the pipe. Others do all of that, and insulate the liner as well, but in your case, that bird seems to have flown.
I do not know what is right for your installation. In my climate, and with a steel chimney chase, (an old air cooled triple wall chimney), I needed all the help I could get to keep the flue warm all the way to the top. Last year, the air cooled triple wall was still air cooling, this year, I have blocked the convection so things will stay warmer around my insulated liner.