zháo reports a result that caught and is now reached (睡著 = managed to fall asleep). zhe (戴著) leaves a result worn and ongoing. The reading splits the two: zháo lands the outcome, zhe holds a state open.
結果補語 names a specific outcome (看懂 = read-and-understood, 吃完 = eat-and-finished). zháo is the one fixed complement for catching, taking hold, succeeding. The result the verb aimed at arrives.
✗ 他睡著在 → ✓ 他睡著了
✗ 他沒睡著著 → ✓ 他沒睡著
✗ 我找了書 (for: I found it) → ✓ 我找著書了
English folds 'fell asleep' / 'managed to find' into the bare verb, so learners leave 著 off and the success goes unmarked; zháo is the syllable that says the try landed.