2nd in command to sauron

[Legolas]

Commading armies is a job any idiot can be given in the real world...being Leiutenant of Barad-dur would indicate that he's not an idiot, and that he could command. Hell, all it takes at minimum is the ability to shout orders audibly.

It's an indication that he's not an idiot, but it's no indication that he could lead an army. If he could, if probably would've been on the war front sooner than that final battle. Armies and such were obviously not his strength.

Also, with that thinking, any of the Nazgul could've done it which Eurytus denies.

Just because The Mouth didn't rule Dol Gulder or Morgul doesn't mean he wouldn't get Isnegard - after all, it is a brand new post! Nobody had it before, there's space for him AND the Nazgul at DG and Minas Morgul to rule!

Given that the Nazgul were used a lot during the War of the Ring and the Mouth was used only at Morannon and seemingly stayed at Barad-dur, it seems more likely that a Nazgul would be given reign over Isengard too, spreading their reach. The Witch-King seems most likely as Sauron would want his most trusted or powerful servant ruling in this, his most eastern tower.