The poskim i quoted above, nosei keilim, seem to say unanimously that one who davened only as shat”z must say elokai netzur, or perhaps according to the nagen avrohom, a variant of a short supplication. Maybe nust yehiyu leratzon would suffice? I don’t know, but you can’t dismiss the nosei keilil with just the idea that you don’t see people do it when they’re doing something which on its face is an ad-hoc practice… was this done in a shul with a posek there at the time? Or just people davening minchah in a shul?
But in either case, taking over as shat”z after the first shatz davened shtil has yet to be shown as permissible… based on the above (did anyone here read my summation of the poskim? Or am i wasting my words?) It seems problematic, and if someone only davened hoich then they’d have to say something afterwards, the rationale being that one must first take leave of the Shechina – a regular shatz does this during the shtiller SE, and takes his leave of the Shechina on behalf of the tzibur by kaddish tiskabel. One who only davens hoich needs to take leave of the Shechina for himself, and after SE is the only time he can do that.
As for why it’s not tircha dtzibura(someone here raised this question) my guess is that there’s just no choice… One needs to take leave of the Shechina; it’s a very important part of davening, so what can he do?