Uh obviously you seed teams I'm not suggesting having the two best teams play in the second round and having some random team get to the quarterfinals without having played anyone.
And ya, why wouldn't you want to have Azure Cats play Bigfile Miracle with the winner playing CLG or something? If they beat another professional team in a 3-of-5 then that means they are doing something right and occasionally, the underdog will win. Right now the best players jump to the best teams because there is no chance for new teams to get in and compete for the World Championships without going through a year-long qualifying bid because the team pool is way too low.
Additionally, obviously if the draw was 128 they would have one big-ticket series showed at all times with most series simply played and recorded for those who want to watch, without professional commentary (again, the way tennis works). You would only watch the game Riot deems most fan-friendly but for those who actually care about what is going on in the professional scene rather than simply watch the same 10~ teams play each other again and again with no stakes on the line, this would be an opportunity to see what some of the smaller teams that never get a chance to compete would want to do.
League will not stay interesting if amateur teams are not even given the chance to compete (and don't try to sell the NACS=>promotional tournament or whatever is the relevant stand-in across other regions as a 'chance to compete', because it isn't). Not to mention that the round of 16 would suddenly mean something because it would mean that a team would have earned their spot there by beating three world-class teams in best-of-five series.
Worlds last year were a complete farce and worlds this year will be as well, and all the LCS shit in between is a poor ripoff of professional sports leagues in the United States. People don't get attached to teams in league of legends, they simply want to watch the game played at its best. Things would be so much more interesting if every once in a while a team filled with names I don't know were given the chance to beat the established teams and prove themselves; and it's hard to argue that whatever team comes out of such a tournament would be a worthy world champion, if only because of how much they had to prove themselves against the best in the world at that time, on that patch. As things stand they basically rig the teams to be established professionals teams out of their own handpicked leagues and then have them play like ten meaningless games before finally moving onto a bracket stage of 2-of-3s (although apparently Riot has realized that they need to at least switch to 3-of-5s?).
Expand the draw, play multiple games at once, make the finals actually mean something. Also, hold tournaments more than once a season so that teams aren't just stuck playing teams from their own region every single week...it's unquestionably awful for the level of play in every region (yes, even Korea could stand to play their inferior opponents and could learn something from them).
Imagine holding four of these tournaments every year; having the NA Open, Korean Open, EU Open, and <insert fourth here>. It works great for tennis and golf and I see no reason it wouldn't work for League as well.