Scaladoc: Refactor Contexts

[bracevac]

Having worked on #23607, there is a proliferation of implicit and explicit parameters in scaladoc (e.g., TypeSupport), which informed #23559 and is becoming increasingly untenable as new features get added over time.

Some refactoring is in order. I would suggest taking inspiration from the Scala 3 compiler, and bundle everything in a single Context type.