Quite a few Final Fantasy games allow you to change jobs freely actually, with the exception of FF3 which required Job Points to be able to switch.
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The difference is, old Final Fantasy was solo. Add other players to the mix, and you can't just throw in any gimmick class just because it sounds fun. Solo, you can play around with your style free as you please, come up with any weird setup you want. You're enjoying the game the way you want to enjoy it. But as soon as your enjoyment can limit others', things get toned down.
I mean, don't ninjas get reduced fall damage and increased movement speed as traits? Which, correct me if I'm wrong, are generally pointless at the end of the day?
This has been designed to have all the jobs the same value in a party. If you have a Dragoon which "Dragonkiller" trait that makes them more effective against dragons, then on Coil of Bahamut, which is full of dragons, everyone will get Dragoon, making other classes excluded from than content. This is a multiplayer game, you cannot make one job better than other for certain task. You're playing with other people with their own jobs. On single player game can work, on online games not, since balance is needed. It is the same for another discussion. The elemental damage. The reason we don't have elemental weakness or resistances is beucase that can cause jobs to be useless or very powerful agains a specific enemy. Imagine a boss with resistante 100% to all elements. You cannot get a Black mage or Red Mages, most of their spells will deal 0 damage, making useless that job. Of coruse we can fix it give more skills with no elements, but we enter again in the HW problems and the vast amount of skills. Current traits are treated around a job specific mechanics, which not affect the party composition or enemy you're fighting. From a MMO point of view, the traits you describe is a very bad design idea.
That isn't entirely true. People will still want jobs that add utility, like Ninja. What it would mean though is they will want to have a Dragoon around to support and it'd be the main DPS of that raid. But the point does still stand that giving jobs an edge over certain things isn't gonna happen, since as was stated already it's similar to elements.
Dragoon getting a fancy jump animation would be nice flavour.