html: improve consistency across private helpers

[milmazz]

Most of these changes are similar to the ones I did in #2181, a good summary of the specific changes could be the following:

• Change most of the private helpers to accept the config value as the first argument
• Transform generate_redirect/3 to a private helper, is not referenced elsewhere.
• Sprinkle some Path.join/2 instead of explicit paths.
• introduce the write!/3 private helper as in formatter/markdown.ex

[github-actions]

📦 Docs artifacts are ready: https://github.com/elixir-lang/ex_doc/actions/runs/21811383974/artifacts/5426631544