I think you and others are getting way too caught up with the term "overpay." The Cubs paid what they had to in order to get him to sign. The amount was higher than most expected, more than most people in his situation get, and more than the Twins probably should have paid. We won't know if Chicago overpaid until after the fact.
If you pay what you had to for the market....why is it crazy or overpaying? I never denied that Baker isn't in the class of Sheets or Webb at full health, but what other pitchers are you using to compare? I hear Colby Lewis but he has plenty of red flags, perhaps more than Baker. (And he's two years older) Every player and their situation is unique, so to suggest that there's some body of evidence that says this was an overpayment is just as flimsy. By all means, post who you are comping him to and I'm sure I could poke similar holes to what you did with Sheets and Webb, but I'd be curious to see it.
If people are getting too caught up on overpaying, perhaps you and others shouldn't make it such a central part of your evaluation.