No, look. Japan never was after Midway, it was too far from their logistical hubs and too close to Hawaii for them to ever hold it. Midway was just the flypaper for the Japanese.
Midway island
The Midway atoll is insignificant on most levels. It’s just a small island at the edge of the Hawaiian island chain. It allows the control of northwestern approaches to Hawaii and is in range of WW2 era aircraft, which makes it a threat Americans can’t ignore. If Japanese take Midway, Americans prioritize taking it back, with prejudice. A submarine blockade ensures Japanese can’t build up forces there and the island would have to be evacuated within a few months. This was the Japanese plan all along, they only planned to stay for a few days to destroy the infrastructure.
However they went after Midway not for the island, but after the American response to the Japanese attack. Midway is too close to Hawaii to ignore and they knew the US Navy had to sally out to defend it. Their plan was to quickly overwhelm the defenders, then lie in wait for the inevitable American counterattack, ambush the carriers and destroy them. The Japanese were after Enterprise and Hornet, they knew Saratoga was too far to intervene and Yorktown was damaged at Coral sea. The plan was to take out the two remaining operational carriers and thus prevent American counter-raids that have annoyed the Japanese over the past several months.
After Japanese carriers were destroyed, American carriers pulled back and began to withdraw from the area. Spureance knew his cruisers were no match for Japanese battleships and battlecruisers and his two remaining carriers both had depleted air wings. Almost all his torpedo bombers had been shot down, most dive bombers from Hornet had to ditch at sea, a number of dive bombers from Yorktown went down with the ship and some of dive bombers from Enterprise had been shot down. He still had about one carrier air wing worth of aircraft, but the dive bombers weren’t hugely effective against battleships, you needed torpedo planes.
Once American carriers began to pull back there was nothing the Japanese could do any more. They could still try to take the island and maybe even prevail (or not), but you can’t force evasive carriers to action with battleships. They have too good reconnisance and too good range, they can harass you and you have no recourse. The only smart thing to do at that point was to preserve what was left and withdraw.
But taking the island was a realistic possibility. The battleships can’t force carriers to action, but they can prevent them from interfearing with the operation too much. The question then becomes would the garrison hold against the Japanese assault or not. Even if not though, the Japanese don’t accomplish anything of value and just take horrendous losses for no god reason.
