I should have said "physical conflict" or something along those lines. Obviously, it goes without saying that Kryptonite should not necessarily be a factor in actual conflict and stakes. It's not a reflection on his emotions or anything along those lines. But seriously, would you just prefer that Superman dominates in every physical confrontation from this point forward? Or when he fights super-powered beings like Zod, they just bounce back and forth for 30 mins of a totally even-keeled beat-down until one of them decides to kill the other, as was the case in MOS?
No. But Kryptonite isn't the only possible answer. Kryptonite isn't even an answer at all. Kryptonite is an excuse for not thinking of an answer.
You bring up the fight with Zod. The fight with Zod was terribly written. The fight with Zod was an indulgent slugfest with no point, no arc, and no tension. You know thw answer? Don't write it like that. Write it smart. Don't have them bounce back and forth for 30 minutes in a totally even keeled beat down, have things actually happen. The Avengers didn't need Kryptonite to make any of the fights involving Thor and The Hulk good, neither do Superman movies.
Kryptonite can be a clutch, and it has been used as such many times in the history of Superman. That's for sure. However, if used correctly and sparingly, it can also be used as a tool. It can serve as a gut-check to Superman, in that despite all of his perfection, all of his greatness, a little rock can bring him down to our level...and he might have to find another way out of that trap that doesn't involve flying or smashing his way out. You get the idea.
There are other ways of doing that that involve putting actual thought into the story. We really do not need Kryptonite, I don't understand why people think we do.
The same kind of tropes exist for every superhero. Why should Superman be no different?
No it doesn't. Doesn't exist for The Hulk. Doesn't exist for Thor. Doesn't exist for Wonder Woman. Doesn't exist for most super heroes. With most super heroes, the writers actually just write. It's pretty much just Superman.
Again, man, I have to stress, Consider what are widely thought to be the best Superman stories:
All Star Superman
For All Seasons
Whatever Happened to the Man of Tomorrow
For The Man Who Has Everything
What's So Funny About Truth, Justice, and the American Way?
Kingdom Come
Secret Identity
DC Comics Presents: Superman and Swamp Thing
Lex Luthor: Man of Steel
Birthright
Out of those ten stories, widely considered to be the ten best Superman stories ever, Kryptonite plays a significant role in the way you describe in exactly one of them, Birthright. And it only plays any role at all in two others, Lex Luthor: Man of Steel and Whatever Happened to the man of Tomorrow. In Lex Luthor: Man of Steel that role is incredibly small and could have been cut out entirely, and in Whatever Happened to the Man of Tomorrow Green Kryptonite, which was used to [blackout]kill Krypto[/blackout] could have easily been replaced with a sufficient amount of force, and it's only Gold Kryptonite [blackout]which Superman uses to remove his powers when he decides to retire[/blackout] that plays a role in the plot, and that role is very different from what you're describing.
Basically, we don't need Kryptonite to create a physical conflict. Nine of the ten best Superman stories ever didn't need it for that purpose.