It'd make more sense if Guwon came across them after Jinwoo's family died.
My guess is that something happened with his father that resulted in Jinwoo having to kill him. Don't know what or the reason why since Jinwoo has amnesia and therefore a very unreliable narrator atp. Guwon stumbled across the already injured and amnesiac Jinwoo and only let him live 'cause Jinwoo lied about being a teacher.
It wouldn't make sense for Guwon to make Jinwoo and his father fight to the death only to go back on his word and reevaluate Jinwoo's life after Jinwoo had to kill his father.
For me already he was there because the first chapter starts when Jinwoo opens his eye and he has bandage around his head someone should bandaged his head because he is on the ground like someone ambushed him at back. He doesn't look like he did bandage before passing out. I can tell Guwon wouldn't waste bandage on his head if he didn't find him interesting already and if someone did that they would have taken him. I think he already tried to attack Guwon and beaten to pulp. As I said its a wild guess. It can't be correct.
Guwon didn't have any interest in him before he found him, tho? Guwon was literally seconds away from killing Jinwoo before he found out he was useful. He wasn't even romantically interested in Jinwoo ((and I still don't think he is, he's just a horny monkey that learned how to cum)) till they started fucking.
We don't know the time frame for when events prior to them meeting happened. For all we know Jinwoo could've bandaged himself and just collapsed due to his injuries + fatigue.
It wouldn't make sense for Guwon to:
A. Have people with "no worth" fight to the death for the sake of the victor's survival only to turn around and be like "actually I'm going to kill you anyways unless you can be helpful."
B. Waste "precious" supplies he's literally willing to orphan children over on someone he doesn't know has "worth".
Guwon's whole thing is that he doesn't hesitate to kill people he thinks won't help his town. So it doesn't make sense that he'd waste supplies on someone he knows nothing about nor had any interest in prior to bringing him into town. If Jinwoo had said he was a cashier the series would have ended right then and there lol
I will put my wild guess here maybe comes true or not but I feel like Guwon meet Jinwoo and his dad and suggested something like 'Fight each other and I will take who wins.' I think that's why we see his dad attacking in dreams.
Maybe at a poin he got interested Jinwoo and killed his dad and Jinwoo attacked him because he killed his dad so he is thinks Jinwoo teach him well because he is the only one to rejected his offer and tried to kill him? Also explains why Jinwoo tries to kill him while sleeping but he is totally fine with it.