While it might be interesting, it would also be hard to balance. The basis of the trinity system is that you need a tank to soak damage from the enemies (because healers and DPS die too easily), you need a healer to counteract that damage (because tanks and DPS alone do not have enough recovery) and you need DPS to beat bosses before they enrage and generally make the content go quicker (because tanks and healers do less damage). The content is designed for a fixed amount of each role. With so many different role combinations this would no longer hold. Consider the following (notation is main/sub role):

A DPS/healer would presumably do less damage than a pure DPS but more than a healer, and would heal less than a pure healer. Since dungeons are designed to be done with one full healer, it would be detrimental to replace one DPS with a DPS/healer (less damage -> slower run; extra heals not needed). On the other hand, if the heals from a DPS/healer were enough to beat the dungeon, it would be optimal to drop the full healer and take a DPS/healer instead. Similarly, if a DPS/tank has sufficient damage mitigation to tank the content, it makes no sense to bring a full tank.

How would this be implemented in the DF? Would DPS/healer and DPS/tank be able to take the healer and tank slots? That would mean the content would have to be designed so that the lower healing output and damage mitigation are enough, further deprecating the actual healer and tank roles. I wouldn't be surprised if some people started abandoned duties when they don't get the right classes. Or it could be made so that a healer slot and a DPS slot can be converted to two DPS/healer slots, which would create all kinds of fun confusion in figuring out who does what.

One approach to balancing would be making classes perform approximately equally as long as they have the same combination of roles in any order. In other words DPS/healer = healer/DPS and DPS/tank = tank/DPS. This would avoid the favoritism but it would not increase diversity either. It might even reduce diversity if a tank/healer combination didn't exist - and that combination could easily be overpowered.

A decade ago I played another MMORPG called Anarchy Online. It had 14 different classes and no clear roles, but instead most classes were able to do multiple things with varying degrees of success. Besides tanking, healing and dealing damage, it also had extensive buff and crowd control systems. The end result was that some classes were strongly favored over others. Doctor was the best healer and was always welcome to a team. Enforcer was usually the first choice for a tank, though a skilled Soldier could perform even better. Bureaucrats with their strong CC abilities could trivialize many fights, and did good damage too. Nano-Technicians and Shades excelled in pure damage. A few other classes were okay too and were generally not turned down. But then there were those classes that no one seemed to want. Fixers had HoTs and some CC ability, but only mediocre damage so they mostly ended up used for taxi service thanks to their access to a unique transportation grid. Meta-physicists had some of the most useful buffs in the game, but since they lasted 8 hours people often just got the buffs and ran off without taking the Meta-physicist in the team.

It's possible a solution exists, but I've yet to see any game implement class diversity successfully.