Ya know where the same formulas are used? Bicycle brake cables. Bends are bad. Lube is good
Remember that science as all about creating approximate descriptions of the physical universe, testing these descriptions, and refining them. Very often, one finds an approximation that 'works well enough', and just uses that approximation to 'do stuff'. This is where science becomes engineering
To a very good approximation, friction does not depend upon contact area. Instead friction depends upon the materials involved, and lubricant between them, and the total force pushing the materials together. If you double the contact area, you double the area available to create friction, but halve the force per unit area, net result total friction remains constant. Again, this is an approximation; in the real world increasing the contact area might increase the friction because there are more places that can stick together, or reduce friction because distortion of the objects gets reduced. But for calculating cable pulling force, it is generally 'good enough' to consider only materials and contact force.
For straight horizontal pulls, the contact force is caused by the weight of the cable.
But when you go around bends, additional force is caused by the tension in the cable pulling it against the bend. Consider a 90 degree turn; you have the drag of the cable in the conduit pulling 'west' (for example), and the tugging force pulling 'north', and the cable is getting squashed against the side of the bend with force to the north-west.
The total contact _force_ caused by the bend in this turn depends only upon the total cable tension and the angle of the turn, but not the radius of the curve. This gets us back to the approximation that friction depends only upon the materials and the pressure, but not the area.
If you increase the radius of the curve, then you get less contact _pressure_, and thus less friction per unit area, but more total area, and thus approximately the same total friction.
This approximation clearly falls down on a number of points. For example, it ignores the stiffness of the cable. Clearly, a larger bending radius means less effort in _bending_ the cable, even if it means the same contact pressure and thus the same sliding friction. It also ignores things like distorting the insulation, or high pressure displacing the pulling lubricant. But it also ignores any 'stickiness' in the lubricant, which might actually _increase_ the friction if the radius of the bend increases.
Since a bend causes friction, which means tension in the cable, and the friction in a bend depends upon the total tension, multiple bends creates something of a multiplying effect; each bend will produce friction that depends upon the friction leading up to the bend. This is why cable pulling tension can be different when going in different directions, and why multiple bends is bad; double the bend and get 4x the bend friction. Also because of this multiplying effect, even slight reductions in friction (using lube) will result in major reductions in total pulling force.
-Jon