Thing is NOMISYRRUC you're looking at it from only one direction the IJN's one and not taking into account the USN's reaction.
If the IJN had an active carrier fleet available then the USN is probably not going to go as per OTL with its operation. They would want to force a battle with the IJN who by now are at the short end of the stick in terms of aircraft development and training. Even if the IJN somehow managed to re-pool all its aircraft together this would not happen as they would then be taking aircraft from everywhere along their defensive perimiter, as well as somehow assuming that the IJN kept its air groups together instead of basing them out of land bases where they were mauled to death (see Solomons).
Realistically the IJN could not keep its air groups together and had to spread them around as per OTL with all the effects that incurred. At the Philippine Sea the poorly trained IJN pilots were ripped apart, you've basically got no cadre to train them so unless the IJN withdrew ALL air support for the island defences and concentrated soley on preserving and enlarging its airgroups and training up new pilots this really won't do much and is basically impossible to enforce.
The three Unryu's were nowhere near completion and again getting the aircrews who were trained to land on carriers was a HUGE problem. Also finding the aircraft themselves to do it was an issue, as not every IJN/IJA plane was a carrier one.
And then you've got the USN's reaction.
Lets go with the IJN somehow still having 13 decks available. The USN will be going after them with everything they have and would probably mass their carriers together to do so. They might even delay any ground offensives until they are sure the IJN carrier arm has been defeated. But this would probably require them attacking somewhere that the IJN MUST defend. Lets assume that if the IJN somehow does not come out to play and their entire senior staff are not fired for cowardice etc, then when the USN attacks Iwo Jima forces the IJN to come out to play in a winner takes all gambit that is basically a roll of the dice. This gives them the worst possible place to attack. Because at least in the Phillpines you've got places to hide and terrain, on the approach to Iwo its just open sea. And if the USN knows the IJN is coming out to play (and they probably would due to the Japanese codes being so comprehensively cracked) then you're going to have every scoutplane, seaplane and floatplane the USN has looking EVERYWHERE for them, along with all the subs they can get on station etc.
also you've got the IJN's mindset to consider. Their love of stupidly over complex plans, the cult of the offensive. Getting them to not do the Phillpines Sea battle would require an act of god in itself.
The thing is Steamboy, is that Hasdrubal Barca asked what if No Battle of The Philippine, not I. Therefore you post should be addressed to him, not me.
I agree with your last paragraph, but I have to add half seriously and half sarcastically that Japan did have an god who could act in June 1944.
I didn't say anything about the Japanese altering the way they used their carrier aircraft before June 1944 because they don't. What I did suggest was that they didn't try to defend the Marinas Islands with their fleet or base hundreds of shore based aircraft there. My idea was that they conserve the carrier aircraft they had in June 1944 and carrier qualify the shore based aircrew that were lost in the Philippine Sea IOTL for the defence of something the fleet would have to fight to the death for, like the Philippines, Formosa or the home islands, but not Iwo Jima.
If the Philippine Sea had not happened its very likely that the Americans would still have attacked Formosa in October 1944 as a prelude to an invasion of the Philippines, which would still be planned for November.
The objective of the American attack on Formosa IOTL was to soften up the shore based aircraft and destroy the remaining Japanese aircraft carriers. It's exactly the same objective ITTL, except the Americans would be expecting stronger opposition because more Japanese aircraft carriers remained. The result would be that all the aircraft saved and aircraft carriers saved from destruction in the Philippine Sea would instead be destroyed four months later in a sea battle around Formosa. I wrote that in my first post.
However, I doubt that it would help the Japanese. I think the most likely result is that the aircraft and aircrew not lost at the Philippine Sea IOTL would be lost at Formosa instead for no significant increase in American losses. If the Shokaku, Tahio, Hiyo and Junyo survive this battle unscathed then they get sunk at Leyte Gulf, if there is enough fuel for them to take part.
My second post, outlined Japan's second option, but it was much more likely that they would go for the first.
As for the availability of the first 3 Unryu's, I checked my copy of Conway's before writing the first post and it said that the first 2 were completed in August 1944 and the third in October. IOTL Tahio took part in the Battle of the Philippine Sea although she had only been completed 3 months earlier. Therefore Unryu and Amagi could have been added to the Carrier Decoy Force at Leyte Gulf and IOTL they would have been if their air groups hadn't been destroyed a month earlier in the Formosa battles. Katsuragi had only been in commission for about a month, but as it was a now or never, all or nothing operation she would have been sent too.