Actually not. Attack and ASW carriers can both handle jet aircraft; LHAs and LHDs can’t. Even the three Essex class carriers - Boxer, Princeton and Valley Forge - converted to LPHs retained arresting gear and catapults. It was too difficult and costly to remove.
You were counting ‘ASW’ carriers. Just how is one of those different than and LCA? Because they still had cats? So what if they aren’t using them? And they were too weak to launch the modern jets of the time.
Hate to break it to you but jets fly off our LHAs. “but but those are VTOL”. Who cares? Argentina learned to respect them the hard way.