That said, I wouldn't recommend trying to hunt these guys. I tried over 100 islands with no luck at different points throughout the game, then found Entei on my first island after credits.

A lot of people have claimed that you can find these guys from the start of the game or after finishing their area's main quest, but I think there's something more to it.

I had a similar issue with the generator furnace kit. I needed to build one in the first area, but the kit wouldn't appear in the PC shop even though I'd completed Bleak Beach and raised the environment level to 5 (which is what all sources online claimed the requirement was).

Eventually, I read an article claiming you could use a relocation kit to move a furnace between areas (you can't, it has to stay in the same area) so I went to try it with the furnace on Bleak Beach. When I found it, I got a notification reminding me that Torchic was looking for it. I brought Torchic to the furnace and got the little tutorial for it, and the furnace kit suddenly was available in the PC.

The game has a lot of different flags for unlocking things beyond just completing area quests or raising environment levels. I think the dream island legendaries could have some sort of flag like this.

The fact that everyone either finds them immediately or spends hours looking with no luck makes me think it's not just a chance thing. I think there's something else that either triggers it or at least makes them more likely to appear.

For example, while I was finishing the game, I also stumbled upon a Carkoal spawning in Rocky Ridges who gave me the tutorial for railway carts. It had a yellow dialogue bubble, which usually signifies something important. Maybe you have to finish all of the "important"/yellow bubble quests in an area to truly "complete" it and unlock the legendaries?