Some real groundbreaking analysis there...

It isn't groundbreaking, but this particular bit is a chunk of reasoning that is seen all too rarely:

Apple needs to earn approximately $12 per share to justify its current price (I am assigning a 15 price-to-earnings to this high-quality but mature business). In other words, its earnings need to almost triple.

In a sense, yes, Apple will presumably someday settle into a normal valuation multiple somewhere around that level, since nothing grows faster than the economy forever.
So yes, either the steady-state (real) earnings per share triple by then, or lower (real) prices than current ones can be expected when the valuation multiples inevitably fall to the teens.
This is simple arithmetic which escapes a remarkable number of otherwise rational people.

In the case of Apple I don't think a tripling in earnings per share is impossible.
It may take quite a while, though, especially if it relies too much on share count shrinkage.

Personally, I cap the value Berkshire's Apple position at 21 times earnings when counting my chickens. (earnings adjusted by eyeball when they seem unusually high or low)
That's probably not the correct number, but it gives a better sense of stable progress than the market price does, and it's not insane.
It's an above-average business that deserves an above-average valuation level. It's hard to say what that premium might be, but I have to pick something.
In effect, the current Apple price around $190 includes about $60 I'm not willing to count on.
If they were making $8 a share on trend right now, there would only be about $21 of market price that I'd be unwilling to count on.
I prefer a valuation metric which roughly tracks the progress of the business over time, not the progress of Mr Market's moods.

Jim