Delete (or move) hand-written HTTP clients

[danxmoran]

#21797 removed the CLI code from this repo, but left the hand-written HTTP clients because they're heavily used by integration tests.

• Determine which clients are safe to delete, and which are used by tests
• Delete all clients that are safe
• Consolidate clients used for testing into a single package focused on testing

Item 3 will probably require significant refactoring, because the existing clients use package-private constants for things like HTTP prefixes. We could possible adapt the codegen system used by the new CLI repo to generate HTTP clients here.