And yet, there's ways to make even a trinity-based DPS-fleet system interesting; it just takes work...
Use varying damage. Neverwinter and other tabletop-based MMOs do this innately, with damage being in the format XdY+Bonus range. Games can be even broader than that.
Use accuracy. A chance to miss adds drama (as any XCom player will tell you) and makes a string of successive blows that much sweeter. It's also a natural place to add range or terrain modifiers (if you can be much more certain of damage by funnelling your opponents in close, you're going to do it).
Use critical hits. Vary the critical hit rate based on superior tactics: better positioning (high vs low, on the hill vs off the hill, native territory vs foreign territory), combo rate (Team Fortress 2 uses this to great effect), or having more territory. Have the effects of critical hits vary - maybe by stat, maybe by weapon, maybe by tier. Have several tiers of critical hits, one after the other, that can cause unexpected advantages or disadvantages: IIRC, it was Flyff that had separate critical hit rates for x2, x5, x10, x25, x50, and x100 damage.
Use ability and weapon types. Maybe your crushing weapons can smack evading troops quickly but are easily stopped by shields, piercing archers can get past damage resistance but similarly shoot right through amorphous jellies, cutting blades leaving bleeding wounds that accumulate on HP walls but slide right off evil skeletons, giant's rocks deal devastating critical hits but can be prevented entirely with perception, and fireball spells set regenerating trolls on fire but find it hard to overcome layered hide armor.
Use area of effects. Melee and point-blank auras are always fun. Targeted areas of effects and location area of effects to take down groups. Lines and cones can block off areas. And there's no need to make them static in effect; a smoke bomb that starts huge but fizzles down, an inferno that grows wider but rages out of control and you have two seconds to evade, or a lightning bolt that jumps from foe to foe, can help players shape battlefields.
Speaking of shaping battlefields: Power terrain, or entire areas where foes can't enter. Superhero MMOs are particularly good at this, with Walls of fire, earthen crevasses, lightning clouds that zap any who enter, storm winds that batter fliers, gravity fields that slow movement and debuff range and deal damage for trying to move. If you can tell a pet "Guard this location," that's power terrain.
Make fights take a while, but not too long. If it takes 30 seconds of toe to toe combat to get a warrior, you have time to survey your situation, to see if you can get help, to evaluate your tactics and possibly switch them up - in other words, strategize and give the player something they can reward themselves with. If every fight takes 3 seconds (1.5 seconds when initiated from stealth), combat is initially exciting but soon feels about as involved as Adventure Capitalist.
Have several damage evasion methods, and have damage evasion that is circumstantial. You might simply subtract damage off the top and feel safe wading through a field of low-level minions without pausing to break stride, or you might regenerate so quickly can safely taunt the big foe to take you on mano a mano.
Allow movement to be impinged, so foes have to take you on to get away. "Clipping," of course, is prohibited, but it can be so much more than that; grabbing on to anyone who tries to run, or knocking them up or back or down so they have nothing to do but fly through the air for a while, or strikes that damage someone's movement rate.
Power terrain is good indirect tanking; just like being Fighty Toughguy McMasterson, it helps you control the field of battle to your advantage.
Poison the information your opponent has available. Stealth was the first; perception range, stealth strength, and sense types can be factors. Use fog of war - and have ways to reclaim it, such as darkness fields, smoke bombs, or simply taking out their radar installations. Create illusions to fake the opponent out (Dota 2 does this) or just disguise yourself (Team Fortress 2 does this). City of Heroes had an status effect, Confuse, that would alter the faction others appeared to be.
Give healers more than one thing to heal. Hit points, sure. Healing death, stock in trade. But healing energy? Maple Story teams gotta have their bishop. Status effects? One of the only strong points about the Final Fantasy MMOs are their various status effects and the way players can heal them. Lingering wounds? Multiple tiers of damage that require different abilities and different levels of investure to address are a possibility; Star Wars had both damage that healed quickly and without any special treatment, and damage that healed slowly without retreating to a medic.
Give healers more than one thing to do. Buffs, yeah, but that's also waiting until a button needs to be hit. Maybe they can also do damage. Maybe they are off-tanks that create power terrain. Maybe they are the masters of stealth. Maybe they are controllers as well (if you can't kill them, disable them and kill them later).
Vary the healing rate. Champions Online allows healing powers to critical hit as well, and has both area-effect and single-target healing powers. (Champions Online also has a dedicated healing powerset where all the powers can also deal damage; the holy light heals your allies and smites your enemies, sometimes in the same power activation.)
Make the UI healer-friendly. Most games require you to just monitor the health bar. At least back up the numbers with color or brightness or SOME sort of indicator that gets progressively worse as things advance.
Give fighting a goal beyond the end of the HP bar. If you're trying to capture the flag, if you're trying to take down the tower, if you're trying to secure the fort, even an imminent death can feel like it has meaning in the big picture.