Shouldn't the estimate of the universe's age be higher, not lower, after the attractive strength of gravity is taken into account?
Attractive gravity means a shorter time since the Big Bang, for any given Hubble value. Here's a picture (from lecture notes by Sean Carroll):

The averaged distances and speeds of nearby galaxies give us the slope of the curve at the horizontal position marked &quot;now&quot;. If you assume they've always moved at that speed, you get the scale factor indicated by the dotted line, while if you take into account that they're slowing down, you get the solid curve, which reaches $0$ (intersects the $t$ axis) at a more recent time.
Wilczek is wrong when he says that linear extrapolation gives an age of 20 billion years. It actually gives about 14 billion years, close to the correct value. This is more or less an accident: it's only true during the transition from matter-dominated decelerating expansion to dark-energy-dominated accelerating expansion, which happens to be happening now. In old matter-dominated cosmological models (predating the discovery of dark energy), linear extrapolation gives you about $1.5$ times the correct age, so it's possible that Wilczek got 20 billion by multiplying a modern estimate of the age by a ratio from an obsolete model.