Why? 50 years later we actually have V/STOL aircraft and helicopters that allow such a ship to be useful.
Amongst others because of the reasons Stevep already mentioned.
Another reason he didn't mention was that the Iowa had a complement of 1800 (this is the number for the Gulf War reduced manning, if you add aircraft you're going to need more crew).
A Tarawa class LHA could carry more aircraft and more munitions/supplies for them and only needs 960 crew.
Ofcourse the Iowa still has a pair of turrets to provide NGS in that plan, but the LHA can carry 1900 marines around...
With that additional manning needed, within a few years of having an Iowa activated you'd have paid for an additional Tarawa.
Except the modern use for a BB is NGS - which puts you close to shore, and most likely supporting Marines. Having the ability to carry some of the air component of a Marine force is a good thing. The Iowas had the armor to stand up to many anti-ship missiles, not to mention their countermeasures and the CIWS. Harriers operating from a defensible ship close to shore have longer loiter time, shorter reaction time, and less distance to go if they're damaged.
Iowas served on the gun-line in Korea, Vietnam, Beruit, and Iraq I, putting a lot of ordnance on target and somehow failing to fall victim to enemy action - more than several more modern US ships can say after similar or even smaller conflicts.
I've never seen a reliable budget number for what it would have taken to make the conversion to remove the aft turret and install support for a dozen Harriers, but I'm willing to bet it was less than another Tarawa-class LHA would have been.
Except that to have somewhat decent range further land inward the BB would have to be 30ish KM from shore at most. That is, unless more modern ammunition is developed for this white elephant (which is certainly possible).
If you look at what distance a carrier usually will remain from shore, you're looking at much further away, beyond the horizon.
Technically you're right in sofar that the obsolete armor on a BB would be impervious to a lot any enemy can hurl at it. An Exocet or Harpoon hitting the armored belt of an Iowa would possibly only chip the paint.
However, that armor wouldn't be covering the hangar needed for aircraft on a hybrid carrier/BB.
Not to mention that there's plenty of modern weaponry which can hit a BB in ways it was never designed to be possibly hit. Even a Harpoon or Exocet can heavily damage a battleship that way, let alone a truly large and/or fast missile.
Furthermore you wouldn't need to sink the BB, a mission kill would suffice.
Plenty of important, modern systems/electronics are outside of the armoured citadel.
Lastly there's another problem with having only one or two BB's in the USN. I have been told that in the '80s the following happened:
Nobody in the USN wanted to get stationed on them, since they were a dead end career-wise. Eventually they got crewed - for a larger part then on the rest of the fleet - by the misfits and the uncapable.