Sure, the skill tree approach is a workable one, and has been implemented successfully in several games. I'm most familiar with it from SW:TOR. It has four base classes per faction, each of which splits into two advanced classes at level 10 (you can't progress before choosing one) and each advanced class has three different skill trees. Two trees are DPS trees with different mechanics (such as DoT vs direct damage) and the third is either another DPS tree (for two classes) or a tank or healer tree (for three classes each). Originally they were actual trees but metrics showed that almost everyone picked the same skills with very little variation, so they were simplified into a linear progression. From what I understand WoW has some kind of skill trees too.

It's worth noting that the tank and healer roles are already effectively tank/DPS and healer/DPS. Tanks in particular generate enmity primarily by attacking the enemy. I can't think of any game with pure healers either. This is in part to make all classes viable for soloing - a healer with no damaging abilities or a tank who can only taunt and mitigate but not attack would have no way of defeating enemies.

As you say, skill trees imply that the player must choose their role before queuing for a duty. So a tank is expected to be a tank and not switch to DPS mid-dungeon. So the remaining question is if there should be varying degrees of tradeoff between the roles. I think that would mostly just muddle up things for little benefit. It's better that all tank classes are equally good at tanking and all healer classes are equally good at healing, to avoid the discrimination scenarios described above.

A potential problem with player-selectable skills is also making sure that the combination of skills is viable. For example, Esuna in FFXIV is mandatory in many raids and dungeons, but the old cross-class and role ability systems allowed it to be unselected. I think this was also part of the reason why the skill trees in SW:TOR were simplified. If SE implements a skill tree system for FFXIV, it will likely be similarly straightforward.