Would your company hire part-time developers? Or maybe short-term?

Part time, maybe. Not short term. It takes that much time to get up to speed on a non-trivial existing codebase and process. The value prop for the first couple months is usually just about zero once you take into account existing staff’s time guiding and answering questions.

Maybe for an agency model where everything is green-field or throwaway. Or to work on a proof of concept for something new (tho usually you want to save that work for your in house team, because it helps keep them happy.)

Or someone with deep knowledge in a particular area where our in-house expertise is shallow to solve a very specific problem or design an architecture and teach us what we need to know to understand and maintain it (example: Solr cluster configuration).

A company I consult with does. Either per job or on an ongoing as needed basis. Depending on job's they can pull in people as needed. It works pretty well for them, but as they grow, they will need to bring on a couple of developers full time.