I've played a lot of Gauntlet, and just recently (2 days ago) got into speed running it. That might come across bragging, but it's mainly to give context to the strategies/glitches and results I'm posting. On my third attempt at a NG+ speed run with the wizard I was able to kill Morak in 58:07, a new record using the first strategy I'm going to talk about. The next day, using both strategies I was able to absolutely crush the old record, and kill Morak in 49:56 on my second attempt (the first one soft locked halfway through.) I didn't discover these tricks, but I haven't seen any talk about them here, and the first one seems to be almost unknown.

The Dark Torrent Trick: The first trick is harder to notice, and requires you to have "The Hollow Chapters" weapon equipped to unlock the void school. It uses the "Dark Torrent" spell, which is the cone of fire that slows and slightly damages enemies in front of you. It has no casting delay or cooldown, and you can walk while using it. Most importantly, the first damage tick happens the instant you cast it. By clicking as fast as you can, you can repeatedly damage enemies with it, at about the same rate ice beam would damage a single enemy. The sheer speed with which you can clear rooms of enemies more than makes up for the slower movement speed of the wizard compared to the elf.

The Ice Beam Trick: If you watched the video you may have noticed the other subtle trick I used. The one that shoots a horribly powerful ice beam constantly as I effortlessly kill spawners and rooms full of guys that you would normally skip. This is what puts the wizard into a league of his own. To pull this trick off you're going to need the "Golden Feather" relic equipped. You stand near a piece of food or a potion, cast your ice beam, then use your relic. If the food or potion was in range, you should now be able to walk around with your ice beam constantly casting. You can do this multiple times as you find more food and potions in a level, stacking several powerful ice beams into a single super powerful beam. You can still cast other spells just fine, so this works perfectly with the great AoE clear of the dark torrent trick.

Edit: The strategies mentioned below are outdated and elf is currently used for all 3 of the boss fights.

The other 2 particular things I want to touch on are 2 boss fights: The Sun King, and Morak. For the Sun King fight, you're going to want to actually pick up both of the potions, not use them to make ice beams. The heart has a damage threshold where it becomes invulnerable again, and using just the beams will require you to go through 3 cycles. Instead, when the heart comes out you should damage it down to around 2/3 health, then use your "Arcane Blast" potion ability to drop it below 50%. Rinse and repeat for the second cycle and you should be able to kill it.

In the run I posted, I was able to kill Morak in 3 cycles. I did 10-12 practice runs on Morak before I did the run, and not a single one was 3 cycles, so don't count on that amount of luck every time. I did however kill him in 4 cycles in probably 75% of the attempts I did, and 5 cycles in the rest, so it is still better than most regular attempts I've seen. The first cycle, you fight him like normal, but you want to make sure you don't destroy the food before you can turn it into a beam. The food spawns where the sword is when Morak returns to his invulnerable form, so make sure the sword isn't going to get hit by a spell, and make sure the food isn't going to land directly on you when he changes back. For the second and remaining cycles, I like to go invisible right as he is transforming so I can shoot ice beams at him in peace. Just make sure the beam isn't passing directly over the sword, as it'll destroy the food when he changes back.

That's about it for my hot wizard tips. There may be a level or 2 where you could get away with using boots of speed instead of the golden feather, but I tested 5 or 6 levels and only a single one of them was slightly (1 second) faster with the boots. You'd lose more time than that swapping back and forth. Maybe someone will find some new exploits, but until then I don't see anyone knocking the wizard off the top of the scoreboard.