In the OP's link, I strongly believe that if those horses were using saddle chariots, it wouldn't have been anywhere near as awful and dangerous an incident.
If the horses in the clip had been using saddle chariots they wouldn't have been in a show ring!!
I'm just not seeing the attraction of these saddle chariots - where and how do you practice the release and step off bit so that in the event of a horse bolting
you can deal with it effectively? IMO the safest thing to do is have a pony properly trained and have a nimble groom who can assist at the first signs of trouble. My husband has twice been on a carriage when the horse took fright at something and bolted (not our horse or carriage) and both times he stayed on and held the carriage from tipping until the driver got the horse safely back under control. Both times the horse involved went on to drive calmly again - if it had been released from the vehicle it would have learnt that when you run, you get away from the carriage - as it was the horses learnt that the carriage came with them and stopped when they did, no problem. I think they are horrible things and are encouraging bad practice - no-one should drive on their own, under any circumstances! Especially people with mobility issues!!
Can you imagine what would happen when your pony gets startled out on the road or on a lane and you loosen it from the vehicle and let go the reins??? It really doesn't bare thinking about.
If my pony spooks (as he can occasionally as he is a very fit and sharp competition pony) I shorten my reins and bring him back under control pronto while my groom helps stabilise the carriage - I don't leap off and let him get on with it! And even if he was bolting blindly I would like to think I was responsible enough to take some kind of action - supposing it was turning him into a hedge but I would have to be in dire straights before I would consider jumping off.
IMO if you feel you need to have a saddle chariot for safety then you probably need some decent lessons instead. One day these things will be banned and they'll all be left sitting gathering dust in garages, no matter how nice a guy the designer is.
As for 'grounding' a horse - it maybe wasn't the best way to stop the horse in the clip but it worked and the horses involved in that drama were all reported to be unhurt.