I thought so too until I saw the BBC special. While it's true that the collapse of the South Tower was not anywhere near as destructive to WTC 7 as the North Tower collapse was, the extent of the damage to the south side of WTC 7 was quite startling; all the windows on the lower floors were blown into the building. And I think the fall of the South Tower did cut off the power in the area generally, although I am not positive on that one.
The
NIST thing (pages 14 - 15) said that the power and phones were still working in 7 after 2 collapsed.
Here's their summary of the damage, perhaps I should just quote it.
Debris Damage from WTC 2
After WTC 2 collapsed:
Some south face glass broken at lower floors
Dust covered lobby areas at floors 1 and 3
Power on in building, phones working
No fires observed
Debris Damage from WTC 1
After WTC 1 collapsed:
Heavy debris on Vesey Street and WTC 7 Promenade
No heavy debris observed in lobby area, white dust coating
SW Corner Damage – floors 8 to 18
South face damage between two exterior columns - roof level
down 5 to 10 floors, extent not known
South Face Damage –
• middle 1/4 -1/3 width south face, 10th floor to ground
• large debris hole near center around 14th floor
• 1/4 width south face, above 5th floor, atrium glass intact
• 8th / 9th floor from inside, visible south wall gone with more damage to west, 2 elevator cars dislodged into elevator lobby
