How does DC transmission require several times less wire than AC?
There are three main factors at work that make DC power transmission more efficient.

Peak voltage.  As you say, AC transmission lines have to be designed for the peak voltage, but the useful power is related to the RMS voltage.  These differ by the square root of 2 since they are are sines.  Power is proportional to the square of the voltage.  The same transmission line can carry twice the power at a steady DC voltage that is the same as the peak AC voltage.

Skin effect.  With AC, the outer edges of the cable carry more of the current.  This makes the cable appear to have a higher resistance at AC than it does at DC.  I don't know what this ratio is at 60 Hz for typical transmission lines.  Of course those transmission lines were sized with this in mind.  This is why sometimes you see a cluster of 3 cables instead of one bigger one.  With DC the one bigger one would work fine, which is presumably less expensive.  There is another issue of keeping the field strength in the surrounding air down, so three lines isn't necessarily only to reduce skin effect.

Radiation and capacitive loss.  Even 60 Hz radiates, and there is always some capacitive coupling to ground and between the conductors.  Radiation is pure power loss, and reactive impedance causes currents in the wires that don't transfer power but cause loss and eat into your max current budget.  I don't know how big either of these effects are with normal transmission lines.  I do know that AC lines are crossed over at regular intervals of some miles.  This makes them essentially twisted pair to reduce the net external field.



7x sounds rather high to me.  Without a clear accounting of what factors were considered and some justification for the numbers chosen, I'm not ready to believe it.  Some amount of extra efficiency in the transmission system is definitely there though.  Note that while DC is more efficient to transmit, AC is easier and more efficient to convert between different voltages and currents.  The system needs to be designed for the best total end to end result.

DC systems do exist, so that fact that most transmission lines are AC probably means that economics doesn't favor DC except in niche situations.  It seems (I don't have direct knowledge) that DC is used today for long distances and when going between different grids.  In the first case, the better transmission efficiency dominates at long distances, and in the second case DC doesn't require the ends to be phase synced.

One example of the former case is the Hydro Quebec transmission to New England.  That's a long haul, but there is also a large power conversion station at the receiving end not too far from my house.  It's definitely not trivial to receive the DC power and make it usable to the local grid.  You can see this for yourself at 42.57047&deg;N 71.52434&deg;W.  The cut thru the forest towards the northwest is where the DC lines come in.  That cut also includes a 3 phase AC line which was previously existing before the Hydro Quebec project.  The east-west lines just south of that plant is a major AC transmission line of the local grid.