the United States uses AV-8B equipped amphibious ships in a similar fashion to the way it used its smaller carriers (eg the Essex class) during the Cold War
Nope. The Essexes either embarked a full CAG (albeit without A6 and F4, chocka with A1, A4 and F8) and did strike ops - and plenty worked Yankee station during Vietnam, or they designated them CVS, filled the deck with S2 Trackers, a few E1 Tracers and used them to provide the ASW cover for carrier groups in 2nd, 6th and 7th Fleet AORs. They never sent them as a substitute for the post Forrestal ships in any meaningful way.
given that the battleships were reactivated for Viet Nam (see USS New JErsey's fun and games there) for Lebanon (80's) and for Desert Storm. Oddly enough, that didn't put the Harriers to bed
While New Jersey
was reactivated specifically to provide NGS off Vietnam, the subsequent reactivation of the entire class in the 80s was to provide Surface Action Groups to counter perceived threat from the Sov Kirov class. They were obviously able to provide NGS for both Lebanon and DS, but they were not reactivated specifically for those ops.
Otherwise, what Orca & Green Knight said.