If we're talking Dyna, one significant downside for DRG is that there is a real threat of wyvern dying on wave bosses, or the occasional wave 2/3 NM or something like an unlucky Mijin Gakure, which *could* leave you gimped for a good bit of time unless you can get a Random Deal/Wild Card. Not something a [pick other DD job] has to worry about.
That being said, DRG's player survivability sure helps overall performance, as Saevel and others mentioned. And it's not just about High Jump and not pulling hate; I'd say another major consideration is that DRG can use a top tier TP set that also happens to feature a ton of DT-. Arguably, Hjarrandi head/body & Sulevia +2 hands are already BiS offensively (depending on buffs/weapon/etc), and choosing Sulevia +2 legs is a fairly minimal offensive loss versus your other options (Valorous, Pteroslaver) - well worth the trade for DT-7% in my book, in any event where you're concerned about potential damage. That's DT-34% already, plus whatever you have on cape (DT-5% or PDT-10%). If you're willing to do something like give up a ring slot, you can fight in near-capped DT- at a rather minor cost to your offensive performance.
I personally play more SAM, and while Kendatsuba+1 gear has its perks too because of high Meva, I don't get the luxury of nearly capped DT- in a top tier offensive TP set. And although DRK WAR can use the same Hjarrandi/Sulevia pieces DRG can, they have some additional options for "better" offensive gear so I find it's often the case that people simply don't use the more defensive-leaning sets, or they just forget (and die).
But aside from all that...
You guys are really doing Dyna with people who would insist on a WAR DRK SAM MNK over a DRG? For Dyna purposes, any heavy melee DD fills that role just fine. Yeah, there are some differences, but it's ultimately a very similar role with similar performance and sorta ludicrous to say any one of those jobs is an inferior option to the point of not being OK to take on the run. I get it if someone has to come support or tank or healer... but if there's a DPS slot available, you're playing with f'ing weirdos if they are insisting that a high end DRG is an unacceptable substitute for a high end WAR or DRK. But maybe this is just some sort of Asura pick-up Dyna party thing that I can't wrap my head around...