I guess that in our society in the western hemisphere, most animals that are being kept by humans, are not "useful animals" anymore. It is not necessary in our modern society that an animal either be a producer of nutrition or a plodder in order to have any value for its owner. Therefore, the question what an American Miniature Horse is "good" for is about as legitimate as asking, what your dog, cat, guinea pig or hamster is "good" for. Winston Churchill put it as follows: "The outside of a horse is good for the inside of a man."
Of course, the American Miniature Horse is not a riding horse, but it is my belief and experience that for many riders - especially children - it is the contact with horses that is the fascination of the horse and not so much its aptitude to be a piece of sports equipment.

Maintaining a miniature horse costs about one-tenth that of maintaining a large horse. The feeding program is basically the same but clean-up detail is much smaller. Because of its small size, it is possible to keep a miniature horse in a good sized backyard so that the rather substantial costs for boarding can be omitted.
Equipment such as halters, blankets, carts, harnesses etc. at the moment mostly have to be imported from the USA to Europe, but first stores within Europe can be found already, too.
Apart from the costs mentioned above, you will also need a farrier to trim your little horse's hooves and a veterinarian for vaccinations and deworming. Additional costs are the fees for the registries/organisations/clubs you are a member of.

The AMHA uses the slogan: "The American Miniature Horse - The horse for everyone!" And it is true. Because of their gentle, affectionate nature and small size, a miniature horse is an excellent family pet. Many people buy miniatures who have big horses and want their children to love, to know how to care for, and to learn to ride years before they are ready for even the most gentle big horse. As the child outgrows riding the mini, it can be trained to pleasure drive. They easily pull a cart or small wagon with children and/ or adults. Older people find that these little horses bring and uncommon joy to their lives and often a renewed purpose and sense of self, not to mention that the grandchildren seem to visit even more. Also, shows are being organized, where the owner can exhibit his or her little horse in halter classes, but also in performance classes, such as jumping or driving. There's a saying in the USA which goes: "They are like potatoe chips - you can't just have one!"