Nope -- with good reason.
1. The thickness of coating he applied, was on the order of microns -- I guess you could actually figure it out from the time*temp it was baked at (if we had this detail), the color achieved (the brass looks pretty light, maybe in the 30-40% Zn range at the optical surface*), and the diffusion rate of Zn and Cu.
*Which is in the top, oh, not quite 100nm or so. That's still thousands of atoms, so the concentration within that layer can still vary quite a bit! The reflected color would correspond to, roughly, the average composition in that layer, I think.
2. ENIG, as the name suggests, isn't just gold. The nickel is put there as a diffusion barrier -- for whatever reason (I don't know, actually; I should see if there's research on why), nickel diffuses very slowly in neighboring metals (Cu, Fe, etc.). This makes it fantastic as a barrier or buffer layer, between a base metal and plating which diffuse together too readily otherwise (e.g, Sn, Zn, Ag, Pd, Au and others in Cu). It's also nearly inert, making it great on top of base metals that are too reactive to plate directly (e.g., Zn, Al) -- assuming you can get the nickel on there successfully in the first place, of course.
Incidentally, Sn and Cu diffuse reasonably well (as also seen in the video, sort of -- the result is a bit patchy), it's just that we can afford to put on a heavy layer, and we slather it with more during soldering anyway, so it's not much of a problem. The interface layer does contain brittle intermetallics, so you do want to avoid cooking it for too long (at soldering temperatures, that'd be... hours+?) which would result in that layer growing in thickness, and therefore weakening the material, and the solder joint.
3. So because of the barrier layer, the gold dissolves quite easily into the solder, at least at typical thicknesses and soldering temp profiles. You do want to have enough solder and time to ensure it's dissolved, otherwise the Sn-Au intermetallics are particularly weak.
Nickel does dissolve in molten solder, but not nearly as fast, so it would take quite a long time to expose the copper directly to solder. Which I suppose is nice because it would eat through the copper foil in...about as much time (~hours)??, assuming there's enough solder present and it didn't have much Cu dissolved in it to begin with of course.
Metallurgy is cool stuff, chemistry with shiny solids and liquids (and rarely gasses*).
*Well, not the shiny part (gasses are mostly clear), but, notable low boiling points include Zn, Mg, Mn, alkali metals and others. Interesting to note there are a number of important alloys that are handled near or above the boiling point of their main addition -- namely, brass (and most bronzes) having zinc (boils 907C) but poured at ~1050C, and wear-resistant steels having manganese (2061C) but poured at ~1700C. Magnesium (boils 1091C) is also used in small amounts with ductile iron (~1500C), and notable amounts (and also zinc) with aluminum (poured around 700C). All of these have notable vapor pressures when molten; but just like alcohol in cooking, it doesn't actually evaporate all that quickly, so you can quite easily finish a process without changing the alloy much. Keep that in mind next time you make, say, a pumpkin pie with a couple shots of bourbon mixed in -- a slice of that is enough to feel a slight buzz.
Tim