ORIGINAL: el cid again
A principle taught in USN is that "no carrier escorted by a battleship was ever lost." This applies to Japanese carriers and others, not just our own. The heavy AA guns of an escort should apply to the defense - even if not the light ones - and it is very likely you will get MORE THAN ONE escort firing if the number of escorts is large enough - in any navy.
The KB at Midway was escorted by 2 battleships - the 4 KB carriers were all rendered into flaming wrecks by air attack and scuttled. The Japanese battleships in the escort contributed nothing of note (that is to say, shot down any attacking aircraft) to KB's anemic AAA defense.
The BB Nagato was in the screen of CV Hiyo at Philippine Sea. The Nagato may have contributed somewhat to the AAA defense of Cardiv 2 but by then the IJN had adopted a much more compact formation for AAA defense (copying the basic formation used by the USN from the beginning of the war).
The IJN's carrier decoy force included two old BBs converted to semi-carriers. By that point in the war they possessed a substantial number of AAA and again the IJN employed a defensive formation similar to the USN one however all four CV/CVLs were either sunk or crippled by air attack.
The USS Princeton was screened by two BBs. A lone bomber was able to penetrate the screen and score a critical hit on the Princeton which ultimately caused the ship to sink.
The number of exceptions to this "principle" seems excessive to the number required "to prove the rule" to me.
Take a look again - the tiny size of the strike that hit Yorktown the second time should never have delivered at all. In any later battle it would not have done. But at that time the fighters were probably less effective than AA guns in USN.
It's my impression that 16 (IJN a/c shot down by CV CAP at Midway) is a larger number than 4 (IJN a/c shot down by AAA at Midway). Of course I attended school before the 'new math' came into educational vogue[;)]. However, you are right that strikes such as the ones Hiryu launched at the Yorktwon would most likely have never made it through the CAP in 1944.