I think there are many things to consider for handling a situation like this.
This is hard, but every show needs to think ahead and have an outline of what they will do in the event of a fatality during a show. Even an accident that leaves a rider in serious, maybe life-jeopardizing condition. Hopefully they never need it. But this is hard to figure out on the fly, with so much raw emotion for everyone.
Do other competitors and participants know? Often if they were not there for an accident, they won’t know if there is no announcement.
What did the spectators see, what was obvious to them?
Does the show have screens to shield visibility to a horse or rider that is down and needs in-place medical attention? Every show should have these and know where they are.
There could be situations where the rider went to the hospital without the show staff being aware of a fatality. Do show staff routinely follow up on a transport, do they ask to be advised of how things go at the hospital?
Who should be notified by show officials beyond the family and trainer? Should the announcer say anything over the loudspeaker? Etc.
I’m sure there are some guidelines somewhere in the USEF rules & regs. I wonder where they are?