Indeed, Kirk is supposedly a war hero, that is, a great military commander ("Whom Gods Destroy"). Couldn't happen before or after the mid-2250s, so DSC is just perfect there.
Kor has also heard of Kirk. He becomes excited when he realizes it's Kirk, when the Organians blow his cover in "Errand of Mercy". So the reputation had to come from
somewhere...
link
.
.
.
AYELBORNE: What are you going to do with him, Commander?
KOR: What is always done with spies and saboteurs. He will be killed, after he has had first-hand experience of our mind scanner.
AYELBORNE: There's no need to use your machine on him, commander. I can tell you his name. It is Captain James T. Kirk.
KIRK: Ayelborne!
KOR: What? (A big grin spreads across his face) Captain of the USS Enterprise. A starship commander. And his first officer?
I had hoped to meet you in battle, but
AYELBORNE: For some reason, he feels that he must destroy you, Commander, just as you feel you must destroy him.
KOR: That's going to be rather difficult now. Isn't it, Captain? What an admirable people. Do you always betray your friends?
AYELBORNE: I didn't want you to harm him. I'm sorry, Captain. It was for the best. No harm would come of it.
KIRK: I'm used to the idea of dying, but I have no desire to die for the likes of you.
KOR: I don't blame you, Captain. Lock up the Vulcan. Take the Captain to my office. We'll have a talk before I do what must be done.
[Kor's office]
KOR: You'll have a drink with me, Captain?
KIRK: No, thank you.
KOR: I assure you it isn't drugged. With our mind-scanner, we have no need for such crude methods.
KIRK: What do you want from me?
KOR: Oh, a very great deal, but first I want to talk. Just talk.
KIRK: You think I'm going to sit here and just talk with the enemy?
KOR: You'll talk. Either here, now, voluntarily, or under our mind-scanner. The fact is, Captain, I have a great admiration for your Starfleet. A remarkable instrument.
And I must confess to a certain admiration for you.
.
.
.
Kirk must have had
some type of accomplishments that would make Kor respect him as an adversary. The simplest explanation is that Kirk fought them before. And something stood out to make Kor take notice. You don't need
Discovery to see that. It's right there, even without it.
But, like I said earlier in the thread, I still don't think another version of DSC happened in the Kelvin Timeline. And I throw anything after 2233 that would've happened in the Prime Timeline out the window.