That's an interesting theory but I don't think I find it plausible. Say we're cutting bars like you said. With the obvious strategy I have to cut the three bars into a total of 9 pieces of sizes no less than 1/3 bar each: I cut two of the bars into pieces of 3+3+1 and one bar into pieces of 3+2+2. Then I give five of the people the size-3 bars, and the other two people each get 2+1.

The two people getting the 1/7 + 2/7 pairs can easily verify they are not getting shortchanged, simply by putting their next to one of the 3/7 bars to make sure they add up to the right length.

(Someone dividing 7 sacks of grain among 3 people can do something similar. Maybe they compare two shares of grain on a balance.)

But if you're trying to give everyone a 1/4 bar, a 1/7 bar and a 1/28 bar, sure, it's “trivially obvious to be fair” if you believe you can divide a 1/4 bar into seven exactly equal pieces. But you can't, some will be a little bigger and some will be a little smaller. Seriously, have you ever tried to cut something us unmanageable as a metal bar into seven equal pieces?