See, threads like this and some of the replies are part of the reason horse people have a reputation for being stuck up snobs!
OP - all horses need to have basic 'flatwork' training before they can move on to things like jumping and cross country. It's the basis of everything (or should be) as it teaches the horse how to move correctly in order to accommodate the rider and be balanced and supple.
Dressage (especially at this level) is that but to the highest level, horses don't need to be trained to the level you're seeing at the Olympics to do other disciplines, but they do need the basics and more depending on what level they compete at.
Dressage is a way of showing your horses training and how well you've 'schooled' the horse and is the first element in eventing because you need to show the horse is well trained and responsive to you, and although fit and raring to go, also well trained and responsive enough to listen to the rider.
At Olympic level, dressage is very technical and precise, requiring the best of training to the highest level, but also a horse that is physically capable of the stamina and movement required and with a trainable temperament - they're often bred for those attributes.
The riders also have years of training to enable them to ride and train these horses correctly and effectively, and they need to have a core and thighs of steel and of course know which cues (aids) to give the horse, in enough time for them to respond to perform a movement.
As someone else said upthread, the less they look like they're doing the better rider they are because they have trained their horse to respond to the tiniest of aids. Me with my windmill arms and flappy legs wouldn't stay on long at all because I'd probably scare and confuse the poor animal!
The movements are all movements that horses can (and do) perform of their own accord, but depending on their training and agility, how 'well' they perform them on their own is different for each horse.
Put one of those Olympic horses next to my old cob and it's like putting Darcy Bussell next to Onslo from Keeping up Appearances - still physically able to perform the moves but obviously one is going to do it better than the other because of training, build and natural aptitude.
On any horse, if you want to go where you want to go, at the speed you want to go at, you need to do more than just sit there, you need to sit correctly and be balanced and give the right aids at the right time. Not to mention sit up and rely on your core muscles if something becomes extremely exciting and they have a good excited jump around!