Yes, nets can be used to catch birds and aren't harmful when done correctly.
Catching a bird whether by using a net or the hand is always a stressful situation.
Given the fact you are relatively new to pet birds and you already have a considerably sized flock (3 budgies, 2 cockatiels), it's best that you slow down for now, take the time to familiarize yourself more with them, in order to prevent an overwhelming situation from happening in the future.
RIP sweet Tito (Summer 2008 - January 17th 2013).
You are missed and never will be forgotten.