You seem very single minded
Two kids fighting over a toys
There's a choice - play nicely or lose the toy
The consequence for not playing nicely is loss of the toy
They have been spoken to calmly and consistently and they know the consequence of behaving badly
Time out, kids getting angry or fighting or in a temper, timeout is a clam place to clam down, it's not a punishment it's a time to calm and regroup
Shouting, sometimes you need to shout to be heard over noisy kids, or immediate danger and shouldn't be ruled out completely (and I'm talkimg about shouting about the situation not at the child) a sharp NO if they are putting a hand in the fire or something
Kids often ply up when there's another adult in the house, probably because mums attention is taken away and they want constant attention, that's normal.
Sometimes when people are busy there needs to be a quick end to a situation, otherwise things can get out of hand and worse, this is why schools issue comsequences - kids distrusting lessons - they are removed to calm and let others learn, detention is iasually for time wasting or homework catch-up a,
There's a time and place for these things, don't dismiss them