Actually ando you may be on to it.There is a spray on grass that can be used & will grow with proper watering.It does work.It's seeds are in a green solution.
Personally i would leave it as is & just water the ground every day since there'll be seeds in the ground.May even get a couple of p r i ckles come up.
However if they're somewhat vain about their lawn not looking green then digging & laying is the best way
As for future mowing best to mow on highest height with the highest mower you have & get a couple of don't walk on the grass signs
edit:the swear meter in this forum is dam rediculous