This is pretty much correct. But in their culture the horse is a working part of making a living, not a pampered, luxury good. Maybe a working analogy would be comparing the life of a stock protector (Pyr, Kuvaz, Anatolian, etc.) or military/police working dog to that of a "lap dog."
It goes beyond that. When I say I grew up amongst them, that means we had had a dairy farm in between a couple of their farms. When I moved from OH to PA and lived near another Amish community, things were not any better, regarding the buggy horses.
There were some Amish families who treated their buggy horses with respect but, for the most part, they were (and most likely still are) nothing more than lowest-on-the-totem-pole pieces of replaceable livestock.
Once they literally got "driven hard and put away wet", you could always find them at the next auction. In my area, as an adult, it got to where they were running out of their typical Standardbred stock and they started going after the ASB's that came thru the sale barns.
Sorry but having lived amongst two different sects of Amish in my liftetime and having seen what they do to those buggy horses, comparing the Amish buggy horse mindset to military/police working dog is an insult to the Military.
When I still lived in PA, there was a Sect about 15 miles from me that had a horrendous outbreak of Strangles. Not only did they refuse to involve the vets in our county, they refused to report any of the incidents. Just let the horses die and they would quietly bury them, in the hopes nobody would find out. They didn't count on the Yankee farmer with field glasses that was watching their every move
That was the year I had to get Strangles vaccines for my horses - it was the brand new one that Ft. Dodge had just come out with and the horse in my avatar still got pretty sick from the new and improved version.
The Amish that broke my friend's SSH brothers BEFORE she bought them caused one to be blind in one eye and the other one's tongue was so nearly severed off, the vet marveled that it healed closed. It left a huge amount of scar tissue and he has to wear a special bit. THIS happened in Tennessee
Again, in all fairness, not all Amish treat their horses that way --- just the majority of them. I love the baked goods the women produce but I have no use for the men and how they treat their animals or their women.