Well, even assuming that skill can be quantified linearly (which is a premise of rankings in general, so it's acceptable here), only the top n results of an n-elim tournament are guaranteed to be accurate. Which is to say, assuming linear skill, the winner is guaranteed to be better than everybody else and the second place player is guaranteed to be better than everybody except the winner. However, in 2-elim, there is no guarantee that the third place player is better than every player in the lower spots (however, it is guaranteed that they are at least better than a certain number of people related to the logarithm of the number of entrants). So anyway we all know that, and it's considered acceptable to give money to the third place person anyway. (Partly because we have pools.)
But the farther you go down, the less accurate the results are guaranteed to be, so the logic of breaking places with matches like that doesn't really make sense mathematically (which is to say, an n-th place player isn't necessarily worse than an m-th place player, where m>n, and the proposed tie breaking scheme doesn't pair off n-place player with m-place players, only n-place players with n-place players). However, the "tie breakers" would offer additional information in the sense that they are matches where somebody wins and somebody loses, and under the premise of linear skills indicate that one person is better than the other one.
I guess the reason not to do it is that since they aren't really tournament matches, it's not really any different from just playing friendlies -- they aren't really establishing the tournament order anyway (which we only really know for the first two players). But they still do provide additional information. Just my initial thoughts.