Many of the constructions performed by Archimedes were neusis constructions.
Some sources suggest that Oenopides of Chios (c. $440$ BCE) was the first to rank compass and straightedge constructions above neusis constructions in importance.
It may have been Hippocrates of Chios (c. $430$ BCE) who publicised the principle to use a compass and straightedge construction in preference to a neusis construction whenever possible.
Euclid's The Elements uses compass and straightedge constructions exclusively.
Under Plato, neusis was considered a last resort when all other methods of construction had failed.
By the time of Pappus of Alexandria, to use a neusis construction when other methods would work was considered a serious error.