Stoves with large amount of coal..

Because the rising warm(ed) air doesn't stay warm. Balloonists continually have to re-warm the air inside of a balloon at a rate that offsets the heat loss of the surface of the balloon or it will return to the ground.
Without the balloon, the air is subject to adiabatic processes:

Too long didn't watch ?

 

To put it in non nerd terms, it is partly just a play on syntax. Heat does not rise, heated fluids rise in a column of similar fluid. Air is, in physics terms, a fluid and thus rises when heated.
It could be referred to as physics teachers being anal with new students.

 

A very large portion of the heat that comes from burning coals is infrared, and that transfers best through air (or a vacuum but most people do not have that situation....) rather than through the walls / top of a stove or insert. Tremendous difference in the amount of heat radiated through clean glass as compared with even slightly dirty or hazed glass too.

When running a stove pretty hard, they can build up a ridiculous amount of coals especially if burning some types of wood such as oak. The cure for me has been to introduce a small amount of under the grate so the coals burn as the wood burns and they do not build up in the first place. Others use various methods to burn down coals and of course they work also.

Brian