I want you to look at this picture. This photo is of a horse in a fast extended trot. As you can see easily here that there is a moment in this gait where not so much as a single foot is touching the ground. A millisecond later one foot will touch and the opposite foot diagonally will also, and then as those two leave the ground again the opposite diagonals will make contact rear foot landing first.
Besides that, at the pace this horse is going he wont be able to keep this up for more than a few minutes without getting winded. Once his respiration is fast soon after sweating will commence. In that respect their bodies work just like ours do. Note that this horse is pulling a light weight two wheeled cart over grass on what appears to be a perfectly flat surface so the owner is operating in a reasonably safe and humane a way.
Lets change this scenario to pavement, and a thousand pound carriage hooked to this horse, with four to eight hundred pounds of passengers plus the weight of the driver. Now lets throw in a not so perfectly flat road. Now lets assume this will be repeated every 20-30 minutes for a period of approximately 3 to 4 hours for a distance of 1/2 to 3/4 of a mile.
Now lets visit what is wrong with that compared to the photo above.
First off I probably don't really need to tell you that anytime a horse is moving fast enough to have all four feet in suspension he is at far more risk of a possible stumble and fall that he is at a flat footed walk. Thats pretty elementary. Should the horse above stumble and fall in the grass he has a very good chance of recovering from that fall injury free. On the pavement that chance is exactly the opposite. Chances are on payment the horse is going to suffer at least nasty scratches, likely bloody road rash, and at worst permanent injury to his knee joints.
Now lets visit some other big differences. This horse is pulling a light weight cart. The added effort to pull it is as little as the harnessed horse enjoys. When we add a commercial carriage onto the horse we have a whole different story. The weight of the carriage alone is at about four times the weight of one of these, then throw in two to four passengers and a driver and now you have doubled that to eight to ten times heavier. Throw in a few inclines and now you have physics adding insult to injury. Lastly the difference of a whole commercial "shift" of doing this verses being "taken for a spin" as the horse's owner above was no doubt doing.
So now you have an idea why just about every major city that has horse drawn carriage rides on their city street have rules and regulations on how the horses can be worked, how many hours, and at what speeds, just to name a few. At least the ones that have their stuff together have comprehensive ordinances. For instance in New York city the horses well being is covered in over 140 rules and regulations. Most major cities follow CONA ( Carriage Operators of North America) guidelines, and the ones that don't frankly should. CONA contends that commercial carriage horses should be worked at a walk, and never faster than a slow trot. *note: the slow trotting is usually for a very short distance for the purpose of getting through a traffic light. Never are commercial carriage horses to be trotted in the manner above.
I probably don't have to point out that most people that want to take a carriage ride aren't doing so to get it over with as fast as possible and most people feel bad boarding the carriage with a profusely sweating horse with his flanks heaving trying to catch it's breath. At least the ones that care enough to notice do. Worked at a walk this isn't even an issue. Thats why the industry standard guidelines are what they are. If you see an outfit that is ignoring these horse friendly common sense guidelines (for whatever their motive's may be) they are probably not the ones you want to patronize.