FWIW, I don't think the Twins actually carried a third catcher all that much. I remember looking at it, and it was primarily at the beginning of 2005 and 2009, when Mauer was coming off injury. (And when September rosters expanded, of course.)
The Twins of that era just rarely spent anything on external upgrades. They had room in the rotation and went with Ramon Ortiz, Sidney Ponson, and Livan Hernandez. They had room at DH and went with Rondell White (twice!). They didn't even make modest additions on the level of Pavano, Cabrera, Crede, Rauch, etc. until TR stepped aside and they got close to Target Field opening.
Keep in mind, Mauer was the most obvious lifetime Twin ever, and they didn't even manage to sign him to any kind of long-term contract until after he won a batting title, and didn't manage to buy more than 1 FA year until after he won a MVP. That's major frugality/conservatism, right there.
I could see the argument that they valued having DHs who could contribute at other positions, but that leads to Shannon Stewart, Rondell White, and later Jason Kubel. Or going back further, Matt LeCroy, and dumping Ortiz with Mientkiewicz at 1B and Morneau in the wings. I don't think having Mauer at catcher had much to do with that preference.