As heating sets up currents the problem with pot configuration is not that the heat can't get to the sides but that once it gets there it is lost through the sides. The larger the surface area the more heat passes through the surface and the less is available to heat the water. It should be clear that the ideal situation would be one in which the surface area of the volume of water were the absolute minimum required to contain the volume. For this to be the case the 'pot' would need to be a sphere - not a very practical configuration for a home brewer. Practicality dictates that we use cylindrical HLTs, kettles, mashtuns.... For a cylindrical vessel the height of the liquid should be the same as the diameter of the vessel. This results in a minimum surface area for the volume contained.
The attached graph shows the ratio of the surface area of a cylindrical volume to the surface area of a sphere of equal volume as a function of the ratio of the diameter of the pot to the height to which it is filled. As the plot shows if the height of fill is equal to the diameter the surface area is only 14.5% greater than that of the ideal spherical container and, all else being equal one would lose 14.5% more heat from a kettle filled to the height of its diameter than from the sphere. If, conversely, one filled to 3 times the diameter or 1/3 the diameter the surface area would be 29% greater than that of the equivalent sphere or twice that of a cylindrical kettle filled to the diameter. Loss in this case would be twice the loss of the optimally filled cylindrical vessel.
Thus a tall skinny kettle or a short squat one isn't as efficient as a 'square' one but it takes quite a bit of 'aspect ratio' to appreciably change the surface area ratio. A 3:1 kettle loses twice as much heat as a 1:1 kettle. If the heat loss is 2% of the total in the 1:1 kettle it is 4% in the 3:1 kettle. Not a big deal. But if the loss is 40% in the 1:1 kettle it is 80% in the 3:1 and it is.
If you want to do an interesting experiment run your kettles as is and then repeat with them wrapped in a quilt or blanket. Be sure the lid is on and is insulated too. Try it with the lid on and off.
The attached graph shows the ratio of the surface area of a cylindrical volume to the surface area of a sphere of equal volume as a function of the ratio of the diameter of the pot to the height to which it is filled. As the plot shows if the height of fill is equal to the diameter the surface area is only 14.5% greater than that of the ideal spherical container and, all else being equal one would lose 14.5% more heat from a kettle filled to the height of its diameter than from the sphere. If, conversely, one filled to 3 times the diameter or 1/3 the diameter the surface area would be 29% greater than that of the equivalent sphere or twice that of a cylindrical kettle filled to the diameter. Loss in this case would be twice the loss of the optimally filled cylindrical vessel.
Thus a tall skinny kettle or a short squat one isn't as efficient as a 'square' one but it takes quite a bit of 'aspect ratio' to appreciably change the surface area ratio. A 3:1 kettle loses twice as much heat as a 1:1 kettle. If the heat loss is 2% of the total in the 1:1 kettle it is 4% in the 3:1 kettle. Not a big deal. But if the loss is 40% in the 1:1 kettle it is 80% in the 3:1 and it is.
If you want to do an interesting experiment run your kettles as is and then repeat with them wrapped in a quilt or blanket. Be sure the lid is on and is insulated too. Try it with the lid on and off.