Ironically, I think that a close approximation to this scenario is found in Monster Hunter: World.
"Tanking" is done by, whomever has the aggro of the monster, whomever has the most life or whomever is in a good position to actually do something about it.
"Healing" is done by, whomever has the items available if they have the time and space to do so.
Meanwhile, everyone is doing relevant damage.
In addition, you can tune yourself towards a particular support role outside of combat if you wish, for example, if you want to Tank more often, you can pick a weaponset that has a shield (Sword and Shield, Charged Blade, Lance or Gunlance) and equip armour that helps you stay in the thick of things (Increased health, Stagger protection, improved blocking etc). Meanwhile, if you want to Heal more often, you can pick a Bowgun that can use Healing Bullets and carry more items that provide healing, cleansing or buffing to allies.
To top it off, all these weapon specializations also have benefits for dealing damage to boot. Sword and Shield has a good combo that enables mounting to eventually knock down a monster to enable big damage but slow activation attacks, Charged Blade can turn into a giant electrified axe that smashes giant fissures into the Earth and whatever poor monster is in your way, Lance can be aimed in several directions making it easier to hit weak spots, Gunlance can explode (Like a Gunblade) which can deal damage through armoured areas and Bowguns have a plethora of different ammunition types that can target monster weaknesses, can apply various status effects to disable monsters or can just dish out massive damage if you use them in their optimum range.
So it's not like your gearing towards a support role leaves you in a state where you can ONLY do that thing. Everyone can still do everything, gearing towards a role merely affects how often you'll be in a position to perform that role.
Heck, for the Behemoth fight as part of the FFXIV Crossover event, they even implemented enmity to further define how often someone could "Tank" (Basically, hit the giant angry Behemoth in the face and he'll focus on you)