What would be a good example for your senario?
Largely comes down to setup and number of players.
MB or RNG strats, nobody is giving you melee buffs. Sure you have frailty, but you're going to struggle to break 5% on the parse. Give your mage a break, focus on hate, chill on the epeen.. it's not going to cut any real amount of time off your run.
Melee strats with 7-9 players, it's not going to matter. You've got your DPS party with DD DD BRD GEO COR WHM, and then probably something like RUN [GEO] [MAGE]. You're not getting the bulk of the buffs unless you force them to rotate for just you. It's wasted effort for minimal improvement.
Melee strategies with 6 players is where it really shines, as Afania mentioned already in another thread. RUN DD GEO BRD COR WHM.. you might be contributing 20% of your groups damage. That's making a real difference and well worth having the set.
How often you see any of these situations is going to be based on who you play with and your/their attitudes toward the game. For some people, it might be 80% of their gameplay. For others, it's likely less than 20%. Outright claiming that 'you should be doing damage' is questionable, because it really doesn't apply to most situations and most people.
If you're strictly looking at best setup for a given event:
Omen - Good place to DD tank, until you have your AF1+3. Then, you job change to something that benefits from cards or you're wasting runs.
Gaes Fete - SMNs. Shut up and G/R/OS. You're doing nothing.
Ambuscade - SMNs most months. Shut up and G/R/OS. Some months, it's a good place to DD tank.
Unity - The only one that requires a tank to begin with is Tumult Curator, and even then it's mostly optional. Optimal setup is SMNs, waste of time to give you DPS buffs and won't make a huge difference in killspeed even if you have them.
Dynamis D - Pulling statues for the BLUs to aoe. Once mobs are slept, you should be off getting more not playing with your toys.
Dynamis D Midboss - Possibly worth meleeing, but dies so fast it's pretty irrelevant either way.
Dynamis D Endboss - Most of these are offensive enough that you would need to prioritize survival over everything. You may be able to swing hybrid on some besides sandoria, but unless you're 6man it's adding extra risk for little extra benefit.
Master Trial: Meleeing is a severe liability in Black and White, but encouraged in the other two. 6man event, so almost certainly practical to do so if going RUN.
Now, I realize the game would be incredibly boring if everyone only applied these optimal strategies. It's good that they don't. But, it's severely lacking in logic to demand a 'MAX EFFICIENCY ONLY' setup out of anything besides the actual most efficient setups.
Groups use varying setups and only a handful of them benefit noticably from the tank's damage. If you're trying to be the best RUN you can be, you certainly should have a set like Taint described. However, consider your own circumstances and priorities.. it may not be the best use of your time to acquire things that won't have much use for you.