The dot product involves projection, extraction of components of vectors, involves things that need to be parallel... the quantity it produces is a scalar and thus orientation-independent. It doesn't have a direction. Work is a scalar quantity. It concerns how much force is used to move an object in the direction of the force (hence, the projection). Dot product is actually very ubiquitous, most of the relationships between quantities you know involve some kind of work with vector's components. In fact, this connection to projection runs very deeply: it works in any number in dimensions, and even in abstract spaces, such as spaces of functions and spectra... the concept of projection is needed everywhere.
On the other hand, the cross product is antisymmetric - it is sensitive to orinetation of the space. It involves chiral phenomena, circulation & rotation, inherently three-dimensional phenomena that use up all three spatial dimensions. It's about perpendicular things instead of parallel. Its result is a vector - something orientation dependent. Torque involves rotation: it distinguishes clockwise and counterclockwise. It has an axis, perpendicular to the plane of motion, it has a direction... the force and the distance to the axis must be non-parallel if you want a torque at all. Similar examples here are everything involving magnetic field (here, the consequences are very deep and fundamental, all the way back to Maxwell's equations).