I've always admired the idea of us having to defend our championship title after gaining.
Logically, it would make sense. If it were real life, we would have to do just that if we would like to remain champion, and if we refuse, we would likely find ourselves being stripped after some time, and title would be vacant--anyone would be able to grab it if they are ready. Would definitely enjoy seeing GameFreak implementing these aspects, and I'm sure others would feel the same!
As far as starting our own gym and whatnot, that would be quite fun, too. Personally, I know I'd definitely have fun running one; taking on trainer challenges, handing-out badges, reward money, and so forth. If a feature like this were to exist, one type-group I would likely consider specializing in is fire, since a lot of my favorite Pokémon are categorized in it (both appearance-wise and competitively).