Mud wasn't trained until she was 3 years old, when I got her, and she's fantastic...but, if I had had her as a puppy, she would be eons ahead of where she is now. I started the boys as soon as I got them and I don't regret it a bit. That's like saying, "Well, my son needs to enjoy childhood and just let him be a kid. So he's not going to start school until he's 16."
If you applied this to horses, you'd be screwed. Horses kind of have a shut off point, unlike dogs...past 5-6 years old, training gets pretty tough. It is entirely possible, but they just won't be as responsive and predictable as a horse trained right from the start. Every natural horse trainer that you speak to will agree that covering all the bases and groundwork as a foal helps develop a well-rounded horse and make under saddle training much easier. Imprinting, a process that involves exposing the foal to many different noises and the like during the first week of life, is a very controversial horse issue. It's a very touchy, sensitive, and somewhat difficult concept to grasp, and if it is done incorrectly, the horse can easily have problems---creating a disrespectful, pushy horse. BUT, if the process is done correctly and the trainer continues training carefully and correctly, you end up with a horse who is calm, trusting, and predictable. I havev an almost 3 year old filly who we raised from the ground up.
She was imprinted at birth and I have continued working with her pretty much all her life. Horses are prey animals, and scare easily with noises, objects, and many things that you just wouldn't think about. If you desensitize your horses to these things, they learn not to fear them. My filly is more bombproof than any of the aged horses I have ever owned. Even at her very young age, nothing frightens her. She has nerves of steel. We recently started under saddle training(she's very, very small, so I held off until closer to 3 years old so I was certain her knees and back were strong enough for this kind of training), and she has never once bucked. She readily accepted the saddle and is generally just doing fantastic. I am a firm believer in imprinting, and when done correctly, the end result is a horse like her.
All animals, humans included, are programmed to learn at an early stage. Of course, animals learn things throughout their lives, but the youngest years are when they can absorb things best. Interacting with their pack members(family--YOU) is fun for pups. It's a part of puppyhood that they enjoy, provided you make yourself a nice person to be around. Training is fun if you can make it that way. If you can't, then you probably aren't enjoying it either and you probably need to learn a thing or two.