Subject to the severity of the tech delay, we used to move aircraft up the line to cover it then delay the next aircraft's movement. Less of an impact of the punters but was always subject to commercial and engineering input and approval.
Example being an inbound at say 0800 planned for a 1000 departure gets put onto the 0700 delayed flight. When (if) the tech frame comes good, assign it to the 1000 departure.
Spare airframes just sitting around is costly for an airline.