Why don't we call the fermions in the standard model force carriers?

The situation is not symmetric at all: This diagram

describes a force between two fermions, but a diagram such as

just doesn't exist (in the Standard Model).

Fermions can in fact mediate a force between bosons, like in:

Such diagrams are highly suppressed loop diagrams though, and the one above would after renormalization be seen as just one contribution to the resummed 4-boson vertex.

I think that this is more about the historical construction of the theory than about the actual interactions. In a lagrangian, two fields A, B interact when there is a product term of both such as AB. So, I see no real fundamental distinction there, even with more complicated expressions.

But when one introduces the interaction bosons, it's by the mean of gauge theories. Basically we start with "a particle" (a field describing this kind of particle), for which we impose a local symmetry. For the Lagrangian to be invariant, we must introduce new fields, which are those of the interaction bosons. (That construction leads to the term for the photon, the Z, the W's)