I'm trying to find examples where the following perspective on bicategories is developed.
We can define a 2-category as being enriched in Cat, where Cat is treated as a monoidal category using the Cartesian product. It seems like you can treat a bicategory as a weak enrichment, where the necessary associativity diagrams are weakened and replaced with the associator and unitor 2-cells.
This seems to make certain things much simpler, particularly if you wanted to work with a bicategory whose hom-categories were categories with structure and the composition and coherent 2-cells were to respect that structure. That also makes me think that something goes wrong when you take this approach, because I can't seem to find any references where this is developed.