Mel, no, not quite right there about the screens. People nowadays are far less like the old racing crowds who used to see broken legs very visibly and see horses shot directly in front of them, along with the attendant thrashing around.
You've got a number of people now who'd probably be well-meaning, but useless, in trying to rush to 'help' and another type who'd want to take photos and gawp. The screens are primarily to give privacy to the VET, since nowadays they're (usually) less ready with the humane bolt, but may need to spend a bit of time with a stricken horse. It's not helped if some Johnny-on-the-rails is bawling out 'is he all right?' and distracting him. There are sometimes also - if the scene is within running distance - grooms, trainers, or owners in attendance - and they don't want the public yelling out gratuitous advice or snapping the inevitable damned photos. I know I wouldn't want a picture of me in floods of tears, while my horse was down, being videophoned to someone's pal as 'hey, look at this drama!'.
We don't normally invite the public to a yard to watch a horse be shot, or to see the vet euthanise our pet cat or dog - I don't think there's anything to be gained from showing the process to thousands of people, many of whom will be very upset by seeing the process, although there are just as many who will just shrug their shoulders and say 'well, that's racing'. Racing plays a lot more to being a 'family' experience these days, not just for the grizzled old hunting types, and while one shouldn't cosset children from life's realities, I don't see the point in scuffling around for ages with a distressed horse in front of them. I realise that some forumites are more robust about these things than others, but I see no problem in also taking into account the more sensitive souls who aren't from farming, huntin', shootin' and fishin' backgrounds and who will go home sick at heart by viewing a horse in its death throes. It's not coy or twee, I think, to find the damn screens and certainly have the vet to hand in a timely manner.