Also trisecting a line isn’t far off from trisecting an angle, so constructing a pair of angles like 5/16 and 3/8 (=6/16) and then trisecting the chord between

Also trisecting a line isn’t far off from trisecting an angle, so constructing a pair of angles like 5/16 and 3/8 (=6/16) and then trisecting the chord between them would work. Wolfram MathWorld says that "[a]n approximate trisection is described by Steinhaus" (https://mathworld.wolfram.com/AngleTrisection.html https://mathworld.wolfram.com/AngleTrisection.html) and this is basically what I have in mind.