We NEVER, EVER allow this. We work harder at NOT letting them buck that very first saddleing than any other time. Horses buck that first saddleing totally out of fear. That new thing tied on them might as well be a mountain lion in their eyes. If one gradually erases all of the fear, bucking does not become the thing they do when they are fresh or get scared. Some horses buck a time or two and get it out of their systems. Others just get practice and get better and better at it.
While a horse is tied solidly, we tighten a rope around the horse's belly. We use the soft cotton rope that we sack them out with. Then, we put a surcingle on one. We put a small rope around the horse's neck to keep the surcingle from sliding back.
When the horse is comfortable with the surcingle, we are ready for the saddle. With the horse tied in a safe place, we throw on a stock saddle and let the horse stand around all day with it on. We have it snug, but not extra tight. We also have a breast collar on it. We move the saddle around a little, make the horse's hip move back and forth, but do not longe it or move it around any more than that. I may leave a horse saddled several hours at a time for several days. This usually gets them completely comfortable.
The first few times I move a horse forward, I will always put an 'over-check' on them. If it will copy, I will try to put a picture of how this works to keep a horse from bucking.
This is going to be a time for us to disagree on all accounts. What you do works for you, and what I do works for me.
Notice I didn't say anything on the preparation before saddling as this isn't what this thread is about, it's concerning a broke horse.
By the time a horse I'm training is saddled, they're well prepared and have a lot of coping skills, i've also done a load of groundwork and can move their feet where ever i'd like. They're used to a rope anywhere along their barrel, all the way back to their flanks and will move out as relaxed as can be. When I do introduce a saddle, it isn't that big of a deal either, they're used to flapping, they're used to the pressure of a girth, the only thing new is the pressure of the stirrups flapping on their sides. Some will buck because of it, and it isn't a big deal. It's all experimentation on the horses part to see if the pressure will go away, some buck because of a bit of worry about it and some buck out of annoyance. In the grand scheme of things, it doesn't really matter which one. I'll go through my groundwork like normal; I'll work them in the round pen, lunge and do some desensitizing. By the end of their first saddling they're pretty darn comfortable in it. After i'm done working my horse i'll
Even after the first day, all of my horses have the option to buck. They have the choice to try whatever they like really, but if it's a behavior I don't want I put pressure on and discourage it. I rarely even have a horse try to buck in the second saddling because they've already learned that it isn't a very good choice.
I would never introduce something a horse could be worried about when they are tied. When I introduce something new I have my horse haltered with a long lead. They've got the option to be worried and move, that is their instinct. I never try and make a horse stand by restricting all movement, I find a horse will get more quiet when they can make that choice, as when they do make that choice they'll be relaxed.
I'd personally never throw a gadget on my horse to keep them from bucking either, they learn it doesn't do them any good the first saddling and that it's a very poor choice their second if they buck then. One reason is I have never needed a gadget to do that, second, I don't introduce a bit until my horses have been under saddle for weeks, I don't even introduce the concept of using two reins for a while. Using a gadget would be completely counter productive to what I do. Next is that I want a horse to be able to lower their head. After that first saddling it isn't rare to see my horses moving around long and low on their own because they're so relaxed, I wouldn't want to restrict that movement.
When I do put the first ride on a horse, there wouldn't be a need for it either. If I thought I needed a gadget to be safe, I shouldn't be riding the horse in the first place. I'm as comfortable on a horse getting their first ride as any broke horse. Not because i'm a bronc rider, my name is breakable for a reason. They're just prepared well. There is no resistance towards going forward and by that time has rolled around, new things that might be interpreted as scary aren't a big deal, they've learned to relax when new things are introduced.