The Secret Service did not shoot him even though Hinkley had already fired multiple shots.
Your Hinkley analogy would work better if Babbitt was a lone actor like Hinkley.
Byrd had just seen the mob overwhelm all of his compatriots and breach every obstacle meant to stop them.
It would make sense for him to try to subdue her if she was the only threat, but I suspect he felt he was the last line of defense and desperate measures were called for.
We obviously don’t know what he was thinking for sure but I seriously doubt his main concern was protecting himself from Ashli.
No it would make sense to subdue her. Subduing the first through the window would perhaps dissuade the rest from going through the window.
Regardless of anyone else assaulting that window/door she was unarmed and did not constitute a threat to his or anyone else’s life.
And by your reasoning it only made sense to shoot her if he was there after going to stand quads over that dood/window to preclude others from climbing through that window.
Byrd gave no warning, Byrd stepped forward, shot Babbit and retreated and was not seen again.
If Byrd was defending that hall way his actions only make sense if he continued to defend the hallway after shooting Babbit.