Healing and tanking has been handled very differently here than in WoW (which I'll use as an example), and I think the key to making tanking and healing less about dps is making them less overpowered in their own role and having less predictable encounters. Fights could have a degree of randomness regarding damage taken, damage could keep coming between mechanics and healing/tanking toolkits could be more limited and require active usage.
In WoW there are random damage hits on party members, heavier boss autoattacks, boss crits with a bigger impact, short cooldowns and random targets for aoe heals (consider a 10-15 second cooldown for Medica and Medica II hitting only 4 random targets), short cooldowns for basic tank mitigation (consider Inner Beast or The Blackest Night replacing tank stance and having to hit it every 10-15 seconds to survive basic damage) and less powerful tank buster cooldowns (shorter duration, less damage reduction). Tank swaps occur more often and the cooldown on Taunt (Provoke) is shorter. Healing in WoW is much less powerful than here and you actually use your single target heals on people, because aoe healing targets random people or eats a quarter of your MP and you don't have the means to recover MP as well as here. Heals are less powerful and Medica II equivalents are considered "raid cooldowns" (usually a 3 min cooldown) which get coordinated with other healers for certain mechanics. Having a raid wide cooldown on resurrections makes it more punishing to die in raids, so healers have to be on their toes for people's mistakes (small mistakes are often recoverable with some healer attention).
Healers in WoW can do damage but they lack powerful damage tools like Holy. Tanks do a decent amount of damage with their regular rotation (mitigation and damage are not mutually exclusive), but can increase damage with talent choices and gearing. Slowing multiple enemies and kiting them is also a thing which can be done to survive big pulls and make tanking more engaging.
What keeps the roles accessible to less than average players are the lack of near-death damage spikes and the focus on sustained mitigation and healing, and also the not very demanding dungeons and raids at the easiest difficulty. Danger in this game occurs when people aren't topped off in time for a big hitter, but in WoW danger occurs when people get lower and lower on health as the healers can't keep up with incoming (random and unavoidable) damage because they didn't find the balance between mana-heavy strong heals and cheap single target heals. I find that the latter approach can give a gradual increase in difficulty across different content and train people for end game, whereas the former kind of danger is either deadly for the unprepared or totally insignificant (like a dungeon tank buster which can be taken without mitigation and no damage follows it).