If you read the article carefully, he also talks about shorter tunnels which would only go a few miles below the surface. Not necessarily through the core, although I believe the journalist got a little confused himself with the Moscow-Washington bit (716 miles depth != Earth core).
Anyway, it's nice to see he solved the pathfinding, but it's probably not the most challenging part of the problem!
Have a look at a globe. (or google earth or whatever) The direct line between DC and Moscow is surprisingly shallow. The exact number does seem too low by a factor of 1.5-2 though.
The exact number is right. The depth is R(1-cos(theta/2)), where theta is the angular distance between the two cities and R is the radius of the earth. (Draw a cross-section of the earth.) Let d = R*theta be the distance on land; then the depth is
R(1 - cos(d/(2R)). (For short distances this is about d^2/(8R), so the depth varies quadratically with distance, which makes sense. But Moscow to DC isn't that short.) The radius of the earth is 3963 miles, and DC and Moscow are 4850 miles apart; the formula gives 719 miles for the depth of the tunnel, which is close enough to the claimed 716 that I blame rounding errors.
Anyway, it's nice to see he solved the pathfinding, but it's probably not the most challenging part of the problem!