Our riding center growing up was run by a retired military officer from the cavalry.
He always insisted everyone, grooms or riders, paid attention so a horse would NOT get loose.
It was not who the horse was, but that a loose horse would cause everyone trouble, including itself.
It was up to the one on the end of the lead rope/reins to insure they always were paying attention and keeping the horse’s attention on what they were doing.
He would say, in the military, if a soldier of any level would lose a horse, it better had a great excuse, because if not, it was kitchen duty for the next week, including the highest officer.
Kitchen duty was peeling a million potatoes and making a million trips with buckets full of potato peelings to the trash pit.
Our instructor would then make sure everyone, at every time, always knew where, how to handle/tie/lead their horse properly.
He didn’t mind repeating himself a million times, always patiently explaining and demanding everyone did as told.
As he would repeat, it was in the horse’s best interest first that we had to do any we did with the horse properly, every time, every situation.
Yes, there are inevitable accidents, but even those some times are caused by inattention and that was our task, to pay attention, until it became second nature to do things right.
I think yes, you need to try to stay with your horse, but first you need to manage your horse so we are not where we are in trouble and need to stick with it and yes also, if the situation gets dangerous, do let go and try to manage catching horse before real trouble happens because of a loose horse.
Both are right, do try to stay with your horse and if you can’t, do turn loose, don’t die trying!