If you still believe this is impossible, think what would happen if a company on the day of the IPO decided to immediately and unilaterally reverse every transaction. Everybody would get their money back, and the company would simply get its equity back. It's as if the IPO never happened. Clearly not an impossibility. The only difference here is that because some time passed the investors want some extra return on their investment. That's not an insurmountable problem because Apple is rich.
Yeah, this isn't a thing the company can do. The board cannot "unilaterally" take shareholders' shares and give them money in return. They can initiate a buyback, but that means buying shares on the open market, and they'd still be a public company even if everyone sold back their shares. Or they can propose a buyout, but that requires a shareholder vote and typically external money.
The first scenario still results in a public company, just with fewer shares. Neither scenario results in the company owning itself.
If you assume that Apple is going to borrow the 600 billion to buy all its stock, then you are effectively saying that whoever's left holding the "private" company is holding a 600 billion dollar loan. I do not share your optimism that anyone can get a loan large enough to buy Apple.
Apple can't take itself private unto itself, there must be a human owner or owners that own all of Apple stock as a private corporation.
Apple could team up with JP Morgan, Silver Lake Partners, and a dozen other firms to take itself private however, and they along with a likely small class of shareholders would own the private corporation (eg perhaps Tim Cook and Laurene Jobs would would choose to retain their large positions).
There is no possible scenario under which a corporation can exist and continue to exist indefinitely in any of the 50 US states without a human owner somewhere in the chain, even if said owner is buried under a dozen shell companies.