I shall take advantage of that opportunity
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amphibious ship will be able to penetrate that same denial zone
No, NOT the same denial zone - as I said, the benefit of naval seapower is to take advantage of enemy weak points along its coast. I am dubious that any country in our current period can comprehensively monitor, nevermind defend, its entire coastline - especially not the geographic behemoths that are China and Russia. While the task for the defender is made easier by certain locations being more favourable for landings than others, there remain some areas which will inevitably less well-defended than others, and amphibious units can take advantage of these weakpoints. These weakpoints, however, are not likely to be the higher-level objective, which are usually more heavily defended. The point of these peripheral landings is to reduce the number of defences holding your forces at risk as they try to reach that higher-level objective. This is even more relevant today given the longer ranges of shore-based missiles.
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Historical evidence proves beyond a shadow of a doubt that helos are not survivable in an opposed scenario
Agreed, but again, must they always assault where the enemy is at their strongest?
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There are few land based weapons that can seriously harm a battleship
Those antennas for receiving fire coordinates and communicating with those aerial spotters, upon which the ship so depends to be effective amidst the smoke kicked up by the exploding shells, on the other hand...
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This involved ships with 12" guns and smaller that were built in the late 1800's and very early 1900's! That is a universe away from a modern battleship with 16" guns, modern radar fire control backed up by aerial UAV spotter/designator aircraft, Tomahawk missiles, and up to 2 ft of high strength, specialized armor designed to withstand 2000+ lb battleship shells!!!!
Two things to address here:
1) Vulnerability: those four battleships that were taken out of service in a single day fell victim primarily to mines, not shore fire (though the latter certainly caused enough havoc with fires and splinter). I like that you highlighted the mine threat in an earlier post - how does that play out when we call for a battleship to get within 20 miles of the shore to deliver fires? How about coastal submarines? Yes, Iowa can take several underwater hits, but so could HMS Invincible - yet, her mined hull had to be beached outside the Strait and she eventually retired outside the theatre for full repairs. How much underwater damage are you willing to let the ship take before it needs to withdraw? How willing are you to fire those big guns when your structural strength has been compromised? Again, I'm speaking here of the primary operational objective (presuming some sort of major city or installation), where such defensive measures can be put into place well beforehand. The problem with battleships is they must always be within close range of the heavily-defended primary objective, while LHDs do not (necessarily).
2) Effectiveness of fires: HMS Queen Elizabeth was also at the Dardanelles. Her 15" guns, though inferior to the Iowas' 16", were hardly anything to stick one's nose up at! Yet, they, too, failed to cause sufficient damage to silence the forts and howitzers - the latter of which were all that were needed to ensure minesweepers fail in their duty to clear the battleships' way to the primary objective of Istanbul. This is compounded today with the spread of self-propelled howitzers and ASM launchers - the battleship gun has changed little, while area-denial weapons have evolved dramatically in range and lethality.
The Yamato and Musashi were sunk with a couple dozen of what we'd today consider to be "light" bombs and aerial torpedoes. To depend on a weapon system that can only perform its duty after it's well within range of enemy fires is, in my assessment, a poor use of resources. Seapower is ultimately about exercising sea control, not to contest it until the last ship floating in a coastal brawl. The best chance of winning in today's world of highly-lethal and inexpensive weaponry is to avoid being hit in the first place - and an LHD that can land its forces in a poorly-defended area overlooked by the defender has a much higher likelihood of usefulness than a battleship.
Now, if we get those 200 nautical mile-range railguns, on the other hand...