Revert "remove caching at github client layer in favor of httpcli caching"

Created by: indradhanush

Reverts sourcegraph/sourcegraph#33219

We should revert this as the implementation of ListOrganizations is now broken.

See review on original PR: https://github.com/sourcegraph/sourcegraph/pull/33219#pullrequestreview-944322413.

And draft PR with failing test case: https://github.com/sourcegraph/sourcegraph/pull/34004 (see updated description in the PR).

Update

This PR now also includes a second commit to include a new test case for PoC purposes. See this comment for more context.

Ideally this commit should be removed from this PR before merging to keep git history clean and consistent. The purpose of this PR is to revert a potentially buggy PR. Any new improvements should be made in a new PR.

Test Plan

Not applicable, reverts a PR.