You discount nuclear-tipped sea-skimming missiles. They exist. For one example:
http://www.fas.org/man/dod-101/sys/missile/row/moskit.htm
And, you don't have to sink a ship, to take it out of the fight. A "mission kill" happens at a much lower threshold of damage.
The minute you introduce this weapon then you open Pandora's box. The other side is likely to retaliate with strategic nukes as well as tactical ones.
Your objections, while true, apply to any ship in any navy in the world. Additionally, first use of a nuke is generally considered a really, really bad move.