Ex. Catch Groudon, transfer to Pokemon Home, trade it for Kyorge. Would I be able to catch another Groudon?
No
One more note—once you've caught a Legendary Pokémon, you can't catch another one of the same species, even if you encounter it a second time in the Max Lair.
It says you can't catch it again. Even transporting it won't make a difference, as me trading away a Pokemon doesn't mean I never caught it. Source