Okay so when I mentioned a comic book I didn't mean that in a negative way, I thought I saw it mentioned here that there's been a Predator comic so I assumed that may have created some deeper lore. But if we're just going off the first two movies, I think that makes it even a worse argument to start complaining about 'canon' here.
First off the original is a write off. There's nothing really established here that's different from Prey or the 3 initial sequences of this movie. If I recall correctly Arnie escapes the self destruct blast, they get picked up by the helicopter, and that's it. So the argument is more in Predator 2. While I haven't sat down and watched the full movie, I've seen bits and pieces here and there as they used to put up movies on cable TV and I vaguely recall seeing the ending where multiple Predators come out. Watching it again on youtube, yes they come collect the dead body and as an after thought the apparent leader and last to leave pulls out a 'trophy' from his belt and hands it to Danny Glover(?) in respect. Coincidentally it appears to be the same gun the pilot is given in the stadium.
Now I would call that an extremely loose bit of lore that you have to stick to. First off no one cared about Predator 2, and as a series it never really went anywhere until Prey came out which was largely successful because it mirrored the original but from a historic setting. Second what happened in Predator 2 is so easy to write around that it's barely a speed bump. Like for one there could easily be different factions of the species here. Or if I were to propose a broad story arch, I'd suggest rather than kidnapping the Predator's honor the victors by creating and preserving clones of them, Mickey 17 style. Then at some point in the distant future a petty tyrant has gotten hold of a collection and is now using them for sport.
I agree the writing behind the coliseum is kind of weak. The 'you will fight each other to the death, then the victor will get to fight me king Predator!' But as a broader story/series concept, I find the idea of bad ass warriors throughout history banding together to fight Predator in the future to be really good.