1. Most people would prefer doing play throughs but feel forced to do the new xp/legendary run of the week, because the rewards are much higher.
2. The game is just not made to be a PvP game and there is no real other "skill" competition. But games need something like "being the best" at it. It makes everything more interesting!
Add a new modus. You have to complete the whole game, kill at least 20 elite packs and 500 trash mobs. Then there is a reeaaaally nice reward, like one billion xp and three legendarys. It must be worth it! Also introduce four leaderboards: 1man, 2man, 3man 4man. There should also be checkpoints at which ur time is taken, like ten of them. Maybe u should not be allowed to change ur specc while on a run. So if u want to improve ur character: U just play through and have a nice reward and no nerd telling you, that doing cave X one hundred times instead would give u more. If u want to set records: Just run through and compete with the leaderboard. Maybe introduce act-runs like this. This would also be golden content to stream/watch! There will be endless discussions about tactics and setups and speccs and routes. There will be pro-teams setting records and everybody will discuss how they did it and then there will be movies of runs and on and on.
Conclusion: Everybody wins. The gamers get a competition. The progamers get a competition. The farmers get more interesting content to play. The community gets heroes.
Tl;dr: Introduce a leaderboard and huge rewards for completing the game in one go.
I wrote a post earlier about different pvepvp games and no one responded lol. I totally agree there has to be some competitive side to a game or a sort of "who is the best" sort of thing. I like this!
I had had ideas like survival of the fittest (non stop add smashing until you or your team dies/mobs are random award xp and some sort of token to purchase fun things/mob spawns can also spawn old bosses from d2/current bosses)
essentially another idea was yours but shorter. Randomized map with 3-10 levels get to the tenth level and port out. Various monster mobs/bosses. Go against other teams etc.