To be clear, I'm not looking to debate whether this is the best Trek film. Rather, I'm asking why so many people see it as such.
I enjoy TWoK well enough, and certainly it is a good film overall. But consider: it is much more militaristic than any Trek before and more than most Trek since, and relatively violent compared to TOS; there is no exploration of strange new worlds; tonally, it is quite different from most Trek stories. (To be clear, I'm not suggesting that these qualities are required for a "good" Trek film -- I'm just noting a few obvious ways that TWoK is unusual.)
In terms of TOS episodes, TWoK is probably most like a combination of "The City On The Edge Of Forever" and "Balance of Terror" -- which, to be fair, are beloved classic episodes, in part because they are somewhat exceptional compared to the rest of the series. So perhaps that gives us some clue as to why the film is so beloved.
In general, TWoK is ultimately about mortality. For all that the film professes to be about Khan, he really is just an Act of God (in the natural disaster sense), creating an unstoppable force that Kirk must humble himself against. The film is really about Kirk learning to confront death -- heightened by the contrast of the new life of Genesis and in his newly-rediscovered son. And that is something that the film did which was new: able to plumb the depths of Kirk's emotional journey at greater length thanks to the larger screen and the longer format.
But, again... it's a great film, but I don't know that it's obvious to me that Kirk learning to deal with the no-win scenario particularly epitomizes what "Star Trek" is (whatever the hell Star Trek actually "is"). In that respect, The Voyage Home seems like the most obvious candidate -- whatever Star Trek "is", to me TVH "feels" more like it than does The Wrath of Khan.
So, why has TWoK earned such a place of acclaim?
(PS: I could write a similar post about First Contact, whose popularity also confuses me.)
There's almost too many good things to make a coherent summary, so let me list some high points: Khan is a tie back to Space Seed, an original series episode featuring Ricardo Montalban. Also, James Horner's sound track was incredible. Kirk's Explosive Reply and Surprise Attack almost tell the story all by themselves. Then there's the well paced storytelling of the cat and mouse battle between Enterprise and Reliant. Throw in some dark sci-fi elements (possibly Alien inspired) like the ear bugs and the abandoned science station filled with rats and dead bodies. There's the Trekish ethical/moral plot of making Genesis vs using it (not unlike Oppenheimer). The self exploration of a middle aged Kirk who's realized he's getting older and regrets missing out on a family. The no win scenario and the loss of Spock. Spock's death is a powerful scene that plays Vulcan logic against human emotion on the stage of a deep friendship. Generally, the film transforms Kirk into a three dimensional character. Those are a few things.
Edit: spelling
Good take - I think you nailed it.
I would add this:
Ricardo f'in Montalban
The dude nailed it.
His chest should have been in the credits for supporting actor.
Agreed, Ricardo did have an exceptional chest. We could all only be so fortunate.