There are some Amish in my area, but I have never personally dealt with any or any strictly Amish trained horses. I do know people who swear by Amish farriers, leatherworkers, and trainers. Its not unheard of for someone to send a green horse to the Amish to be saddle-broke, and then take care of subsequent training themselves.
While there are certainly cases of poor treatment of Amish horses (not that this makes them "special"... horses are mistreated every day by non-Amish), there is a strong economic motivator for the Amish to take care of their horses and to keep them healthy and strong. Their livelihood depends on "horsepower" in many cases - it just isn't logical to destroy something that essential. To draw a parallel, it would be like a non-Amish person going out to the garage and purposely ramming the family minivan into the wall a few times. Will it be as reliable? No... you might not get to work in the morning.
No, working Amish horses are not pets - they have to earn a living. If they don't, it's a waste. Coming from a farm background myself, I learned early that unproductive livestock that must be supported rather than contributing to the farm don't stay around long, my horses being the major exception. Again, to draw a parallel, its the same as if a pizza delivery company has a fleet of 5 cars, only uses 2, yet legally must pay insurance on the 3 idle cars. Its probably more profitable to sell the 3 idle cars and not have to pay for their isurance and upkeep.
Going back to the question of ninteenth century horsekeeping practices, at that time horsepower was supreme, and most horses did not have the luxurious lifestyles that most enjoy today. Then, as now, there were brutal and benevolent owners, riders, drivers, and grooms, but the horses still had a task that they were expected to perform. If they didn't, the owner effectively threw money out the window - something that the average person could not afford.
The working lifestyle was evident in the shorter useful lifespan of horses. Not too long ago, a 13 year old horse was considered to be nearing the end of his productive life - a 30 year old was quite ancient. Now, horses perform beautifully well into their 20's, and the 30 year mark is becoming less novel. The same extension in life expectancy can be seen in humans as we have moved from a lifetime of hard farm labor to a majority (in developed countries, anyway) holding less physically demanding jobs. Modern medical care plays a role for both as well, no need to discount that.
I'm not condoning ill-treatment by anyone, and sorry for the novel, but this is an issue that I have devoted some thought to in the past, from the perspectives of economics, history, and horsemanship. The Amish livelihood depends upon the horse as a capital good: a means of production - even under the best of care, such use has an effect that is generally not seen in recreational horses.