Yes they are a high value ship in terms of flexibility and capability but in actual cost they are not much more expensive than a frigate and cheaper than a destroyer. The original Escort Cruiser concepts had Sea Slug, later ones Tartar and then Sea Dart, they were designed to be capable of independent operations but intended to actually defend less capable platforms from air and submarine threats, that is why they were proposed to Australia instead of DDGs.Something like a Hyuga class ship would never be deployed alone. It's a high value asset, designed to support & work with smaller ships. Anywhere there might be a significant air threat, it'd be escorted by an air defence sestroyer.
So, thinking of it as an alternative or supplement to frigates, which might be sent where a single destroyer or frigate might be sent, is a mistake IMO. As designed, it's an ASW flotilla leader, & for Australia to have five would necessitate the building of more ships (destroyers at least) to escort them.
LPH? Yes, if you redesign it to have more internal space for troops & their kit, but what do you lose? Or do you make 'em bigger? Sea control? Well, not with fixed-wing aircraft. There's a reason that Cavour & Juan Carlos were built that big: they're sized for F-35B. Ise & Hyuga would be fine with Harriers - just add a ski-jump - but there are no more Harriers, & they'd be marginal with F-35B. So if you're thinking of something smaller, it's limited to helicopters, & won't have much room for troops unless you leave out something else.
I'm not sure I'd want to put such a high-value ship close to a potentially hostile shore unless absolutely unavoidable, & where else would it be needed for MCM? All the likely places I can think of would be better served by specialists, helicopters which could be based on land or smaller ships, or equipment mounted (perhaps in easily removable modules) on much smaller ships, which can be devoted to mine clearing full-time while the big, expensive flat top does something it can do but they can't.
It looks to me like a ship that needs a bigger navy than the RAN.
I posted this here as I thought such a ship could be appropriate for the RN, the navy that seriously considered using light carriers to replace station cruisers, it was this idea, I believe, that led to the Escort Cruiser. At the upper end the concept was a missile cruiser without the Mk-26 6"guns (although a few sketches included Mk-6 4.5" twins) and the lower end was basically an escort carrier with a DDG sensor suit and armament.
My mention of the F-35B in relation to the modern iteration, was at the end of a sentence including Merlins, Wildcats and Crowsnest as well as a few possiblys and maybes, very much a, if you have the money and decide to make it bigger sort of proposal. Look at what has been achieved with Ocean and just imagine a better designed and built vessel, that can do pretty much everything Ocean does but excels at supporting ASW and AEW helos and is perfectly able to defend its self. An RN ship would have Artisan and Sea Ceptor, an RAN ship would have CAEFAR, Vampir NG (ASMD setup) and ESSM, either would be more capable of looking after themselves than a similarly equipped frigate. First they would have a greater radar horizon as the radar would be higher up combined with the same missiles as the frigate, then you have the possibility of Crowsnest (Merlin AEW or what ever it is to be called), more helicopters, potentially a lot more, covering ASW and anti-surface maybe even Apache / Tiger.
Its not going to happen, I know that but I do believe it is worth questioning some preconceived ideas as to what can and cannot operate independently (not that anything should if the threat level is too high). Not yet but I would not be surprised if we eventually see FFH, DDH, DLH? operating instead of traditional combatants. Volume search radars combined with very effective air defence missiles (USN is trailing ESSM in an anti DF-21D role reportedly with considerable success), high energy lasers, helicopter and UCAV, maybe even davit or stern ramp launched UUVs. Modular batteries/containers of Spike ERs, or even GMLRS positioned on sponsons off the flight deck that possibly could even be accessed, removed and replaced by a deck crane.