I used to work at a stable when I was much younger, in my teens. We had nearly 90 or 100 horses and ponies there.
Horses do choose to lay down for a nap. They aren’t necessarily sick when they do it, sometimes they just want to lay down, I guess. (They will also roll around on their back sometimes like oversized dogs, too. Usually just after you cleaned them up or when they’re damp from rain or at some other very inconvenient time that will require you to spend more time grooming them.)
That said, it seems to me that at night horses almost invariably stand up to sleep. When they do lay down it’s typically, as mentioned, during the day for a nap, not an extended slumber, and often there are other horses awake and alert nearby. Also, a horse has to feel pretty relaxed and comfortable and safe to do this - nervous horses stay on their feet. So, if a horse is spending the night in a roomy box stall (for example) and feels very safe and secure he might decide to lay down - but he doesn’t have to.
Also, when healthy horses lay down for a nap they often are off their feet but still hold their heads upright. Laying out flat on the ground, with the neck stretched out and the head on the ground, might be alarming to horse owner. However, horses that are sick usually display some other signs of it besides just laying down. These signs can be subtle, but they do exist, so lying down + unusual sweating + labored breathing is not good, but laying down + breathing easily and calmly + no sweating probably means the horse just wants to lay down for a short nap, nothing wrong with him at all.
I doubt the horses linked to in the picture are laying down due to “extreme fatigue” - a fatigued horse can still sleep standing up. Also, as noted, they’re still holding their heads up. More likely, they’re very well cared for horses that feel safe with their owner nearby and they’re just taking a quick nap in the sunshine. I also note that their ears are a in a relaxed and confident position, which would be less likely in an ill, beaten-down, or mistreated horse. Looking at the other horses on the site, they all seem like well-muscled, well fed draft horses with healthy, glossy coats. There aren’t any close ups, but from what I could see of their feet their hooves are all in good condition. The horses are alert but relaxed. There’s no way to know for sure, but they look like well cared for horses.
I know sometimes people get concerned with horses being worked, but like people, horses benefit from exercise (at the stable I worked at the horses were exercised daily except in extreme weather, late pregnancy, or if there was some illness or injury and they routinely lived into their mid-20s which wouldn’t happen to poorly fed, mistreated, or unhealthy horses) The horses on the linked site are all draft horses, bred for heavy work and enormously powerful animals. He uses four horses to pull his wagon, and he does give them days off.
In sum - I see no reason to believe those horses are in any way mistreated or overworked.
I follow the wagonteamster’s blog regularly. He does seem to take good care of his horses. He usually travels 3 or 4 days and then takes a day off to rest. Glad to hear that it’s not unusual for horses to lie down.