The answer is yes and no. It all depends on the family and the pony, meaning that not every pony is a good fit for every family. Ponies have a reputation for being stubborn and difficult. So before rushing out to purchase the nearest pony you can find, do your homework. Understanding the breed, time, money, and proper pony care routine required will help you make the best decision for your family. Many people make the mistake of thinking that a pony is cheaper than a horse or requires less care than a horse just because it is smaller. WRONG. They require the same vaccinations, hoof trimming, de-worming, feeding, grooming, stall mucking, vetting, training, teeth floating etc. that horses do. The time and money required to care for a pony or any horse are not for the faint of heart.
For families that select the right pony, it becomes a treasured member of the family and lives its days carting children and grand children around the pasture or even the show arena. Some families are not so lucky and the cute little "Christmas Pony" becomes a burden. This can happen for any number of reasons: the children loose interest, the pony is not child friendly, the family doesn't have the time to devote etc. Whatever the reason, the family is faced with the heart wrenching decision of "what to do now?"
Training is an option for difficult ponies and their owners. Many times it's not just the pony who needs a lesson, but also the owners who need to be trained how to communicate with their pony. Bad habits, destructive and dangerous behavior can often be resolved with training and TLC. In situations where the only answer is to re-home the pony, rescues, equine therapy organizations, or friends who have horses are a good place to start. The pony can always be sold to another family or sent to auction. If you are concerned about the pony's welfare once he leaves your care, then use caution when deciding what the next step should be.
Every year there are approximately 100,000 unwanted horses in the U.S. alone, not excluding the "Christmas Pony". Should you find yourself with a pony in need of a new home, educate yourself and be cautious of your options. Not to open the flood gates on this topic, but there is a huge market for horse meat overseas and horses from the U.S. are shipped daily to slaughter houses for consumption. That may or may not be an industry you wish to support, so be aware of who you are re-homing your pony with.