Quest Rogue is powerful, but already it has shown itself to not be unbeatable. They are usually pretty slow in the first 5-6 turns, so if you can keep the pressure up on the board and maintain control of it, you have a good chance of winning. Certain control decks can also deal with them succesfully.
And on the subject of having quest cards, every class has viable decks that don't use quests, many of them even have decks that are very similar except they use less cards to build towards the quest and focus on other things.
If you want to get far in ranked without epics or legendaries, you can do that with a solid budget deck and skills to go with it. As has been mentioned, zoolock and hunter probably have the most success with budget decks.