And what would be some instances where 'standard circumlocutions' used to avoid them?
For example, one can define contractability and compactness (for metrizable spaces) in terms of maps of finite topological spaces and orthogonality of morphisms (see examples in the section on topological spaces in Ncatlab, Lifting property. Perhaps words of Thurston apply to these examples: these reformulations might "lend good insight to a variety of questions" in topology "but that is generally not worth developing in [this] case because there are standard circumlocutions that avoid it", namely these standard circumlocutions are the standard theory of contractability and compactness in any textbook.