As suggested by @ogrisel in #20711 (comment)

In terms of PRs, I think it is possible to migrate all doc building to github actions or azure:

1. Build docs with "doc diff mode" or "full doc mode" on github actions or azure
2. Store the build into an artifact
3. Trigger a circleci workflow using the circleci API with a parameter pointing to the artifact.
4. The circleci workflow will download the artifact and host it, which should be quick.