That's clearly not a real live video what you've linked. (Cool band though, I do reckon they would be up to it.)
It sounds like in the actual recording session, at least the guitar's volume pot was cranked up for that chord, or it was taken from a different take with higher gain setting on the amp, or perhaps the guitar was recorded DI and volume-automated before the amp-sim or re-amping. Of these possibilities, only the volume pot is really usable for live playing. That in fact is usually a pretty good option: not only does it come for free in almost every electric guitar, it also has a particular feature that many guitarists aren't aware of but tends to be quite benefitial. Namely, as a side-effect of electric guitars' rather primitive ancient circuitry design, turning down the volume pot does not just reduce total volume uniformly. It also inhibits the pickup's resonance peak – that is the midrange-boosting effect that gives every pickup its “characteristic sound” and is largely responsible for making the guitar appear loud. Therefore, turning down the volume pot quickly makes a noticeable difference to how loud the guitar is perceived, without however actually reducing the total level and in particular the lower range so much that the guitar would fail to fulfill its accompaniment role.
Good control of the volume pot can allow extremely dynamic playing. Grandmaster of this technique is Jeff Beck.
That, as well as normal playing dynamics, can only come through though if the amp allows it. A hallmark of a good tube amp is how it reacts to dynamics from the guitar: at low input levels, it sounds clean (but not quiet/thin); as you increase the level it becomes crunchy but still round and warm; and only at really high inputs does it actually sound distorted. Ironically, the crucial point is that the amp actually performs a very effective kind of dynamic compression:
- it brings quiet inputs up in volume so you can actually play quiet notes without them getting lost in the mix
- while keeping loud notes in check level-wise (so you can also play loud notes without blowing everybody else away), but still making them stand out as more aggressive thanks to the extra overdrive harmonics.
By comparison, transistor-based circuitry tends to either not compress at all (which makes it impossible to actually play as dynamic as you'd like without disrupting the band sound), or compress so much that the result sounds the same, regardless of your dynamic level.
So, as far as gear goes, I would recommend mostly avoiding effects and instead relying on guitar and amp alone. If you can't afford a good tube amp, a digital simulation should do the trick as well – there are really good ones available today, though especially many older ones actually didn't do the dynamic thing very well.