The short answer is that LAEs are 'supposed' to follow the manufacturer/NAA/Operator's approved data. The problem is that much of this data, including checklists is buried within mountains of superfluous catchall warnings, cautions and instructions. We don't have for example a QAR sheet to follow when powering up the aircraft or running engines.
One airline I used to work for had Engineer's Engine Run Check books. Superb, simple to follow and fool proof. The airline decided that it was too expensive to keep these updated and scrapped them. 🙄
Not following the approved data is breaking the law but sadly, especially on the line during an ever shortening turn round time, needs must. Telling the airline that there will be a delay because the approved data is a bit tricky to follow doesn't go down well. They just want an on time departure. 😕