>>49160876Depends on the region, but it seems GL's can challenge other GL's for badges freely. We don't have concrete evidence for the process outside of Galar where it seems all GL's get a free pass to the champion tournament, but presumably since the Gym Challenge is just a way for trainers to prove themselves strong enough for the league I'd imagine the league would let GL's in since they're already strong enough to lead gyms.
>>49162488From a game design perspective: because bosses are more memorable when they have a gimmick or trait that make them unique in their game. From an enjoyment perspective which you can apply to game design and in world lore I guess: Because giving Gym's easily comprehendible specialties and themes makes for better celebrities. Types are easier to understand and get more invested in than a strategy or no gimmick. You can do a lot with Fire types so a Fire gym is cooler than an "Entry hazard strategy" gym. People are fans of certain types because they like the pokemon and the attributes of it.