Sure, Luffy doesn't need to pass out at the same time as Kaido.
Just apply this same scenario but to a more extreme degree.
If Luffy is still standing, but Kaido isn't, Luffy will be the winner.
If Luffy passes out 2 minutes, 5 minutes or x minutes after, he would still be the winner.
When Kaido recovers from his defeat and rises back up x minutes after, it doesn't take away the fact that Luffy did defeat him.
If you don't like the idea of Luffy remaining passed out to show how dependable his crew is.
You can have Luffy comeback and duo with Zoro, the same way they did in Punk Hazard.
By that point, Luffy would have already achieved his goal.
He would have defeated Kaido 1-on-1; them putting Kaido down for good would be a separate task.
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Defeat = Getting Knocked out.
Did they knock out the Minotaur? Yes, they did.
That's why they correctly believed that they defeated the Minotaur and moved on.
When Bon Clay sees the Minotaur, he questions if they really did knock him out.
They did, we can go back and see, so they did defeat him.
However, Crocodile explains that defeating them doesn't prevent them from getting back up because of awakening.
Luffy can fight Kaido.
Luffy knocks him out.
Luffy is still standing, Kaido isn't.
Luffy is viewed as the winner, no problems here.
Later on, once Luffy has moved on from Kaido because he has correctly assessed that he's defeated him.
He could eventually pass out.
When this is the case, Kaido can rise back up thanks to his awakening.
It doesn't change the fact that he got knocked out and that Luffy beat him.
In fact, Kaido himself could literally say that Luffy defeated him.
But at some point, we're going to need to go beyond knocking him out;