Sugar IS toxic if it hangs out in the blood too long, with adequately high levels. It produces all sorts of bad effects - from AGEs to diminished vitamin C uptake (and thus immunity). Sugar is uniquely toxic to teeth, there is nobody that will argue that.
This ... angry chef ... doesn't actually provide anything but confusion. I highly recommend you to avoid it and do your own reasearch.
I know that's not actually a low carb diet, at least by the standards most people use when they talk about low carb, but it is quite a bit lower in carbs than what the average American eats.
I've been eating this way since September 2015, and even this moderate reduction in sugar has been enough to cause a change in how I react to sugar.
I don't suffer any adverse reaction, such as headaches, if I have something high in sugar. Well, at least nothing I have ever noticed. What has changed is that I just don't find anything particularly attractive about high sugar foods any more.
For example, if someone brings in donuts for the office and I eat one or two it tastes OK, but it is nothing special. It's not really a treat like it would have been a a few years ago. I will now almost always pick a salty snack like potato chips over a sugary snack.
For example, we might now have the 'scientific knowledge' to explain why a particular mushroom is poisonous, but it probably started with a 'avoid it because your cousin died eating that one', followed by the observation that this happened to multiple cousins, followed by some kind of proto-scientific experimentation. None of this might qualify as 'scientists' doing 'science' by our standards, but it's still knowledge.
That doesn't necessarily mean homeopathy should be taken seriously at this point though.