Buildings need proper levels of wall insulation.
As long as outdoor and indoor temperatures are different, heat will pass through the materials that make up the walls. Only insulation can prevent it.
Wall insulation is at least as important as attic, roof and floor insulation. Exterior walls involve a much wider surface, and that counts a lot.
The ideal levels of insulation depend greatly on the climate where you live.
If you live in a cold climate you may need R-40 (Metric U-value: 0.14), that is, about 13 inches/32 cm of fiberglass, or 10 inches/24 cm of mineral wool. In hot climate you need a lot less.
For a Passive House approach you need more (in a cold climate): around R-50 (Metric U-value: 0.11; about: 12 inches/30cm of mineral wool).
Wider studs allow for thicker insulation which is especially important in cold climates.
A 2x6 stud wall allows about 5 1/2 inches of space for insulation - significantly more than 2x4 walls, but a lot less than what can be installed in double 2x4 walls.
The studs in the walls are a source of heat transfer.
Framing elements can seriously degrade the R-value of the walls (about 20% in wood framing and up to 50% in steel framing).
To reduce thermal bridging to a minimum, consider the installation of a few inches of foam or mineral wool all over the outside of the wall.
Buildings, in most climates, should benefit from a continuous layer of insulation on the outside of their external walls (insulative sheathing)
But there are cases where that continuous layer of insulation is placed on the interior of the wall.
Exterior walls do not need very high levels of insulation in hot and tropical climates. 2-3 inches/5-8 cm of a rigid insulation is probably plenty.
If building a new home, do not try to make savings by reducing wall insulation levels. Do it right the first time… Doing it at a later moment will cost you a lot more: retrofit insulation is a lot more expensive.