“why do all shuls (outside of Litvish Yeshos) NOT follow this Chazal and have a proper chazoras haShatz?!”
Incorrect.
Sephardic/Eidot Hamizrach congregations at times do same or similar. I have seen it with my own eyes.
If you would take a break from your Litvak bashing hate agenda and look around a bit, you might see it yourself sometime.