It´s up to you to ignore the given tools and in your special case it´s poor raid design and not an issue with the class, but still playable.
Away from this, E11s is even the most melee friendly fight this tier, if not from the whole expansion. Those 2 GCD´s means nothing there. On the other hand you could go for 2 ranged attacks, it´s still 300 potency and in the end probably more worth than nothing just to save your rotation. It might be even possible to hold the gap as close as possible with the use of the backjump in the safespot combined with a delayed jump back to the boss.
About the class thing... Redmage and Summoner are BEHIND ALL melee classes in damage this tier. BLM is the only number 1 close followed by SAM. And again, it´s only because this tier is really really friendly to all those BLM´s out there.
Overall the balancing is fine, all of them are close to each other and that one rare DPS loss caused by the obvious fact that you´re a melee, is nothing. Only phys. range could debate about the big gap in damage for some reason, even if i think 2k DPS less by top players is still "ok". But it definately shouldn´t be more and might move to 1k.
Melees have their options and you can greed the hell out of a lot of mechanics. SAM can easily use backdash, range, dash. DRG could possibly do the same. NIN has ninjutsus anyway and MNK the SSS or chakra for those 0,5s. Especially in E11s you can reach 100% uptime easily. Even with my MNK it works nearly perfectly, so it shouldn´t be an issue with DRG, who has ~0,5s more on the GCD timer.
And yes, MNK doesn´t have to worry about range and that´s great. SSS is a great tool and fits the class 100%, even if you might lose that 0,5s or whatever if you´re using it in some situations. The DPS loss about that is a joke and it´s definately calculated from SE since, again, the balancing is really good and well played melee careers are top DPS imo. If the mechanics continue to be so "BLM friendly", give him and SAM a little nerf and all is fine.
Agree 100%