Tachnun {being a Supplication} is an extension of the Amidah, and hence should be recited as closely as possible to the Amidah; Being recited after Chazoras haShatz implies that Chazoras haShatz is considered as if everyone is reciting it, and naturally the inverse is:- How serious it is to talk or conduct any other activity during Chazoras haShatz.
This being in addition to a Pesak that my Rosh Yeshiva ZT’L gave me decades ago.