I was going to make this topic myself after some more playing around and now I'm sure that it's actually programmed like that. The auto attack simply stops after the death of any monster, even if the monster was in a party and the next monster is targeted and claimed or even attacking you. There is an additional condition that must be met to begin auto attacking which is that you must start with another action or attempt to try an action of any kind.

Normal conditions for auto attack to begin are...
1) Be in action mode
2) Be close enough in range
3) Be facing the target

Additional conditions for auto attack to begin on a new monster (regardless of claim)
4) Reactivate the action indicator to begin auto attack

I made a post about the action indicator disappearing after the death of a monster here.
http://forum.square-enix.com/ffxiv/t...t-of-a-monster

Basically if you are fighting multiple enemies, don't expect to be able to stand there just targeting them one by one as auto attack continues to beat them down in a efficient manner. It won't do that and all you need to do is try to preform a battle action of some sort to make it continue. (cure, weapon skill, buff, etc) Sadly this system combined with other mechanics as promoting many versus one or one versus one battle while penalizing one versus many combat styles with things like the claiming system and AoE battle, this mechanic is harder to recognize.

There is possibly one possitive to this system which would be under the idea of a crowd control measure. For example if a few melee players were in the middle of a large group of monsters that were slept, it would be horrific for them to continue auto attacking onto another monsters and waking them all up.

In terms of auto attack modes in FF XI, the mode could be seen as turning the /autotarget function off so that the next hostile monster is not engaged automatically when the current one dies. What probably needs to happen is for there to be a way to toggle it on so that people do not stand around thinking auto attack will just start on claimed monsters that are beating them while targeted properly. If you ever played major events where crowd control was a vital strategy, this was an important mechanic to the game which all melee were encouraged to use. In FF XIV though, since we are still in active mode the whole time, it looks very strange that the auto attack is not resuming onto the next monster while conditions seem to be all met. 11 showed very clearly that auto attack was not continuing by always removing the weapon from action mode but 14 does not do such a thing.

I have a similar experience to the OP with a levequest "Where's the Meat" where they spawn 3 goats at a time. The current class I was testing was Marauder and as I killed the first goat, I then stood there stupidly as the next 2 were hitting me while I was facing and targeting them with my axe out. I stood there as my HP dropped from 1700 to 300 while watching myself not attack once in probably 15 seconds of time. Of course I popped a cure to survive and auto attack would resume.

Basically if you want to start hitting the next target, pretend that you have to claim it all over again or just make the action indicator return with a attempted skill use of any kind. Yeah I know it sucks though and needs a lot of work.