One thought about Drew's early UZRs:
If I understand how UZR is measured, a player's UZR can improve from one year to the next simply because the average talent level of the players at that position has declined. The top six shortstops in defensive value in the NL in 2007 were the following:
Omar Vizquel
Troy Tulowitzki (age 22)
J.J. Hardy (age 24)
Jose Reyes (age 24)
Adam Everett
Jimmy Rollins
Also in the mix were Khalil Greene in his young defensive-whiz phase, plus veteran or mid-career glove specialists Rafael Furcal, Alex Gonzalez, Cesar Izturis and Jack Wilson. I would guess you could make a plausible case that that was the best collection of defensive shortstop talent in a single league in baseball history. Most of those guys were back in 2008, but by 2009 Everett, Izturis, and Vizquel were gone, and Reyes and Greene had injury-related dips, thinning out the competition a bit. And that's when Drew's UZR kicked up.
So you have at least three possible factors interacting to create the sudden jump: the inherent imprecision of UZR, the likelihood that Drew gained skill and/or confidence with playing time, and the fact that his performance may have been measured against a somewhat less lofty bar after his first couple of years.