As far as I can tell, the use of "node" as jargon came about right around the time that the concepts of barrel vibration and positive compensation were becoming widely known. At the time, maybe 20-25 years ago, a common misunderstanding was that barrels vibrated in a sine wave, and you could adjust your charge weight to coincide with one of the nodes on the sine wave. That explanation is mostly incorrect.
Today, people seem to get that for the most part, but the terminology has stuck. Today, "node" is generally used to mean "good load", or sometimes pertaining to one specific variable. Like "velocity node", "seating depth node". It's an unfortunate misnomer that causes a lot of confusion.
Adding powder increases velocity on average - always. This is easily provable. I've done it, as have many others. Shoot 30 rounds with varying charges and make a graph. You will see a straight line with all the points within the expected statistical variation.
I think the "every barrel is different" thing is overblown. Today's custom barrels are pretty good, and while you'll see very minor differences, they're basically the same. If you get two barrels from the same vendor and chamber them identically, you'll have very close to the same results. Maybe not by gnat's ass benchrest competition standards, but by most shooters' standards.
The reason for varying velocity hitting at the same vertical is that the barrel whips a very small amount when fired. Depending on when the bullet leaves, it will be sent off on a slightly different trajectory. Sometimes, that slight variation in muzzle movement can compensate for a slight variation in velocity, and you wind up hitting the same point. This is what is known as "positive compensation" these days. It was first observed over 100 years ago, but only managed to filter into common understanding in the last 20 years or so.