I visited the million acre Roaring Springs Ranch in eastern Oregon last year. Their buckaroos are the real deal and can ride anything thus the term Buck a roo. Because of the milage they ride each cowboy does have a string of horses that they are expected to shoe themselves and the shoes don't last long when you are riding as much as fifty miles a day. These guys would probably prefer an unbroke horse to begin with. How long does it take to train a horse when you are riding it 12 hours a day?
Don't they have a Branding competition there in the spring?
To answer your question, from my experience, yes the horse gets ridden for many hours that day, but he only gets ridden once a week. And from what I learned is that when you are spending that many hours on them you don't create more work than you have to to save your horse. So you have to use work to train. You don't go out and lope 25 circles on your way out to make a gather. You would wear your horse out and he would be tired by noon. So you use your resources. Teach them to stop good when your turning cattle back, teach them speed control when your roping. Teach them lateral movements opening and closing gates and rolling cattle over when you have something roped. On our easy days on the way back to the barn or trailer we would all do little training sessions as if we were in an arena. Also if I had a day off and had one up that hadn't been ridden I would ride in the barn lot to work on stuff. But that rarely happened.
In my opinion it takes a while. They get "gentle"(I use that term loosely) enough fast because they have to go to work right away. That's when the speed training comes in. We start the colts fast, but right. They may not have a whole lot of ground work and the best place to be is on them, like I said before.
Another factor is how many guys had been riding the horse. If they are on ranch were they hire a new crew every spring and fall then you have quite few guys riding him with different methods and abilities-that's going to slow the process some.
Also in the winter when just feeding cows, everything gets turned out, and I might only keep one horse up for a month before I switch because there isn't a whole lot to do. So if your smart you would keep your young and tough horses rotating through the winter to keep them rode and get them a little more broke. Keep your broke horses turned out to give them a break.
So it really depends on the situation the horse and the cowboy(s).... it may take just a couple of years or it may take several. I have had some in my string that were in the double digits that I still wouldn't consider trained!
Usually your boss will only rope your colts on day that would be good for them as not to "pull their plug". But it may take a little while to get them really solid in a bridle