lever, pole, crowbar, carry on the shoulder
杠 centers on a rigid bar used to lift, pry, or carry: a lever, crowbar, or pole. The shoulder-carry sense extends naturally from using a pole as a carrying tool.
杠 combines 木 (tree/wood) with 工, which likely contributes the sound and suggests a wooden tool or bar. The exact historical development is uncertain, but the phono-semantic structure is typical.
A wooden 木 bar 工 becomes a lever or pole: picture a sturdy wooden beam used to pry open a heavy lid or carried across the shoulder.
For コウ, imagine using a wooden lever to lift a heavy stone with a loud 'ko' sound as it moves: ko -> コウ, and the wooden pole does the work.
large beam balance; weighbeam