The biggest problem I see is the following situation:

Tank queues into a high level duty such as Midas.
Healer: "Why are you dying so quickly?"
Tank: "I went full DPS on my skill tree."

tank dies to tank buster because lack of defensive skills.

inb4 you say something like "they should let the other guy tank" or, "they should be prepared"

Majority of players that play warrior or something will go with the DPS side of the tree and not care what anyone thinks. And not exclusive to warriors. This could also happen to other roles as well. A white mage lacks big heals because they wanted Stone IV. A DPS doesn't meet the DPS check because they wanted some other bullshit.

They would have to design it in a way where no class would lack any of the skills required to complete the duty.

And I think that's the reason everyone is able to play any class on one character, and why there are so many classes per role. To negate the need for a skill tree.