From Extractors to Pseudorandom Generators?
Salil Vadhan wrote to me that the answer to my question is known, and PRGs are equivalent to extractors.

Quoting him:

"See Proposition 21 and the discussion following it in my survey http://people.seas.harvard.edu/~salil/research/unified-icm.pdf
(There's a typo - "black-box hardness amplifier" should be "black-box PRG construction")

It says extractors are equivalent to black-box PRG constructions where you only care about the amount of advice, and not the running time, in the reduction.  Asking for bounded running time amounts to asking for extractors with "local list-decoding"."