Currently what I’m doing is killing the Chief with Trespasser and then killing Vorac with a DPUH/Grog, Sheriffs badge, a crappy corrosive immunity shield and Legendary Gunzerker COM.
@Handsome_Dad’s video kept stalling for me, but I’ll assume it showed a successful way to deal with him 
On the whole and for most characters, taking out the chief at the start is my preferred tactic as well, and I believe most players of my acquaintance do the same. The exception is Gaige, at least for the tactics I run.
That said, it kinda stands to reason that going through one or several shield cycles is how GBX envisioned the fight. Mind you, back when Vorac was first enticed to leave his comfy pit in one of the nethermost Hells the level cap was 50, and we could do substantial damage to the shield in both phases.
On OP levels, both the chief and the midgets pose enough of a threat in themselves that it’s not worth the hassle to go through the shield cycles, IMO. There’s also the slight problem with less than stellar design - by keeping the chief alive we have more midget spawns and thus more second-wind fodder, but the first thing that happens when they catch a DOT even from his regular attacks is that they run for the hills.
Aside from this, babysitting the chief in his shield-less state is simply too risky when both he and the midgets can easily down you while the chief isn’t tanky enough to withstand even a single stray bullet.
It’s also worth keeping in mind that there’s a reason this raid is considered the most RNG-laden one in the game. Vorac seems to suffer from serious mood swings, where his enraged phase can be filled to overflowing with leaps and bites - or he can be downright lethargic and just spam his novas and the occasional tail spike, with only a token leap now and then to show he could snuff you if he could be bothered.
Take out the chief first thing, is my advice, and hang in there till you get the timing of his novas down. The first one in particular has very little in the way of telegraphing, while the second is consistent in relation to the first.
On a side note, as Sal you might benefit from swapping between a Heartbreaker and a Grog in your off hand since it’ll increase your DPS while providing enough healing to keep you safe.
Leg Gunzerker or Monk is a matter of taste, you can usually stay alive with Harold-Grog swaps during the Monk’s cooldown while potentially (Moneyshot RNG willing) getting much higher DPS.
Getting your Harold-Grog swaps down to second nature is also good practice for fighting him as characters who aren’t blessed with relative immortality.
Maya, for instance, can usually finish this fight quicker, but she is not allowed even one single mistake 