The reason for the barrel is because of the inexpensive nature and ease of placing them. Most individuals building a
rocket mass heater have little to no
experience with masonry, so this is a great way to "cap" the stove with extreme ease. Barrels can be found for free basically anywhere, so that is another reason. You also get a lot of immediate radiant heat off of the barrel, so for those looking to get heat fast, barrels will shed heat quickly and effectively to the room.
You can brick around the riser to make your first bell of the stove. Do more than two inches. Two inches is the minimum, and is not worth the headache of having problems down the road with a restricted airflow. Those with lots of rocket experience have found that you can basically increase the gap between riser and bell to as high as you would like as long as the ISA (internal surface area) of the bell doesn't exceed the maximum for your stove dimensions.
From Batchrocket.eu
Riser diameter / Internal Surface area
12.5 cm (5") ISA 3.7 m² (39.8 sq ft)
15.0 cm (6") ISA 5.3 m² (57 sq ft)
17.5 cm (7") ISA 7.2 m² (77.5 sq ft)
20.0 cm (8") ISA 9.4 m² (101 sq ft)
22.5 cm (9") ISA 11.4 m² (123 sq ft)
25.0 cm (10") ISA 14.7 m² (158 sq ft)
Also, a single skin of brick around the riser has the potential for much leaking. The temperatures of the stove may expand the brick and mortar to the point where you get fractures. This might result in smoking on startup out of the bell, as well as drafting issues if the draw of the stove is coming from many locations around the stove. To do a proper brick bell, you would need to do a double layer with an expansion joint between the two layers. A lot of work in my opinion and easily solved by using a barrel.