I was also watching live when Charlie Kirk died.
I also, back in 2020 (I think), caught the tail end of George Floyd's death being livestreamed.
Both were shocking (mostly because they were unexpected) and I thought about them a lot for days afterwards. But, as @SandSandwich says, I managed not to dissolve into a puddle of trauma, I just got on with life.
I'm pretty non-plussed about this announcement TBH. The GOP knows its audience and is playing to them a blinder with this. I actually think it won't happen in the end (the live streaming I mean) but the press release allows the GOP to posture and craft the type of image it wants for a while.
I don't think the perpetrator deserves any dignity or human rights so, in that sense, I couldn't give a shit if his execution is public.
If you're worried about children seeing this, don't let them see it. Confiscate phones when it's scheduled to happen. Turn off the WiFi. Parent your children for an hour or so.