Now, dual-specs exist for just this reason -- i.e., so you don't have to quest on specs that are really designed for group play -- and I could avoid this problem if I really wanted, but here's the thing: I really like being a tank/healer. Whatever it takes to be a truly competitive DPS, I just don't have it, and I will tank or heal 5-mans and raids, happy as a clam, and hopefully contributing to a lower dungeon queue. By contrast, dailies leave me trying to collect every quest mob in sight to get enough Vengeance to AOE them down efficiently, but it feels really inconsiderate to do this while other players are trying to get the same mobs. And other players are always after them, because everyone's on the same rep grinds. Every day is like being trapped in the starting zone of a new expansion, and I honestly don't know if I have it in me to do this all over again on my alts (who are -- surprise, surprise -- tanks and healers).
People who play tanks and healers feel, I think, especially pressured to advance their rep grinds quickly given a raid's dependence on the quality of their gear. It doesn't help that the Stone Guard, which is probably the first encounter most people will see in the expansion that isn't a world boss, is pretty baldly a tank/healer gear check. So the "daily problem" is a nasty confluence of social pressure, higher mob health pools, being on the low end of gear at the start of an expansion, playing a low-damage spec, and the move to a new model of character advancement that just doesn't play nice with group-oriented specs. Except the game needs tanks and healers, and rightly or wrongly, it feels weird to get "punished" for playing them.
So what to do? Bite the bullet and pay to respec DPS every day, like warriors and paladins had to do back during classic and The Burning Crusade? Beg the guildies to baby-sit me during dailies? Play Rent-A-Thug if they're not around? Or just accept that this is the new normal, and move on to War and Peace?