How many times do we get men on second or third, or both with zero or one outs and we can't advance those runners?
The situational stats at b-r.com for this season don't directly answer your question, but ..
OK, so while I had those pages up, I looked some more, and something popped out at me.
Leading off innings, the Twins' on-base percentage this season has been .292, instead of their overall .313. Across the majors, those numbers are .307 and .316 respectively. A bit of a drop.
Leading off the game, Twins' OBP is even worse, .265, and for the majors it's .319 (you'd indeed expect it to be higher since the first inning is the only one the manager ordinarily exerts direct control over).
It's not just Brian Dozier; his 50 times leading off the game have resulted in just 13 successes, but the 18 PA by other batters have led to 5 successes for only a .277 OBP.
We're now down to small sample size, but that's a deficit of 4 first-inning rallies that were at best deferred until 2 outs remained. Obviously, that's not enough to account for anything actually meaningful over the course of a season - on average a couple of runs or so, going by long-term proability tables. But with this team, I think it's going to be little things like this that add up to the lack of situational offensive punch.