Wizard consistently clears GR80s in 3-4 minutes... with no augments. Wizard can run T13 off the bat with all non-ancient, mediocre gear.
Anyways, this thread is interesting: I didn't think so many people would be willing to champion their class and be so defensive about it. If you want a *real* answer though, you need to clarify a few things.
First, running T13 isn't smart. Streamers only do this to show off, but really efficient players run at most T12 (you can watch the paragon 1500+ seasonal streamers these days running T12, so when you think you can run T13 at paragon 800 fast enough to justify T13 that's your opinion, but you're definitely not being as efficient as you could be).
Second, which class is best depends a lot on power level, your playstyle, and preferences. But there is unanimous agreement among the top players that wizard is the fastest class for speed farming, by a margin. That's why every paragon farming group is running 3 wizards with a support monk. Nothing else. Think about it: if that is so fast on GR80+, why would anyone else run something else? Inspect a few top players and you'll notice that almost all of them have a wizard, even if it's not their main class. That's how good it is.
Close second to the wizard is the monk, and some say it's even as good as the wizard. In T13 clearing competitions, the monk is the closest thing to the wizard; however, the wizard still reigns supreme for a few reasons: 1) The wizard's beam just clears the entire screen, once you have enough Archon stacks and therefore enough damage. Just circle around and everything's dead. 2) The wizard's scaling is just insane. While every other class relies on the typical skill+items+sets+gems multiplicators, the wizard adds the Archon stacks - if you're playing on the right difficulty you should have 100-150 Archon stacks all the time, and with overlap that's 200-300 combined Archon stacks. Everything melts, you don't even need the Manald Heal proc. 3) In the beginning of the rift many classes struggle a bit to "get things going", and that includes the non-MH wizard. But Manald Heal is the solution here: once it procs, the first elite is dead, In-Geom is up, and the party begins. Basically, the wizard has infinite power scaling, and the lower you go the faster it is; if you think it's not fast enough then you're either playing the build wrong, or playing on a difficulty too high (i.e., not gaining enough stacks).
After that follow Necro and DH. DH has two builds that both have issues: Impale one-shots elites but you'll struggle to kill trash. UE is the "most fun" build for many, but you need insane power (paragon 2000+ with full augments) to be able to clear T13, and even then it's questionable. As for the necro, its damage spikes are ridiculous, but it's just too unreliable in certain situations (doors are you mortal enemy, sometimes long cooldowns hold you back, no corpses = no damage in many builds, and pet builds lack AoE compared to monk/wiz). I love Fluffy for all his guides and his compassion for the necro, but on "even ground", i.e., comparing necro to wiz/monk with similar gear level, it doesn't hold a candle. It's still viable enough (if you enjoy it) to farm on the necro if you want.
Crusader is in-between. It's still insanely powerful and fast, and especially the most laid-back class as you just ride through everything and it dies. The problem is that you're either only killing stuff you ride through or have to rely on the ridiculously long Bombardment cooldown. The damage a LoN Bomb crusader can deal every few seconds, wiz/monk dish out multiple times *per second*. If you feel crusader is fast that's good for you! But as numerous competitions have shown us, it's fallen behind. It was once the fastest class, but others have caught up now.
Barb has the usual power struggles. And WD just isn't made for speeds at all. WDs either farm on wizard or pay their duty as "rift closer" in groups.
Wizard > monk > necro = DH > crusader > barb > WD.
Note: this is mostly not my perception or opinion, but that of the top players who have min/maxed everything and who recently discussed this on streams. Some of them would put monk ahead of wiz, but on that I'd disagree simply because of streamer "Baudusau's" competitions which has been won by a wizard the last 438958 times (despite a lot of high-paragon monk participants).