There is kind of a fallacy at work here. Just because the color becomes indistinguishable under water does not mean the line (or lure) becomes invisible.
It still has a certain degree of light reflectivity, and it will appear somewhere between very light and very dark, depending on the amount of its light absorption. Fish can distinguish very well between light and dark, no matter whether they can register the color or not. So a dark red color on a lure, for example, might appear black or dark gray to the fish, but it certainly would not become invisible to the fish.
The same holds for line. The fish might not know the line is red, but it will see the line at some level on the gray scale.
If you really don't want the fish to see the line, forget the colors and use fluorocarbon line. It has the same refractive index as water, and it truly does disappear under water.
If the line or lure is in the upper 4 ft. or so of the water column, then the light hasn't gone through enough water yet to filter out the red color, and it will truly appear red to any fish which is also in the upper 4 ft. So red colors on flies and topwater lures do make sense.