Delete (or move) hand-written HTTP clients

#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.

Start by inventorying the hand-written HTTP clients and searching the integration tests for their use. Done means deleting clients that are safe to remove and consolidating test-used clients into one testing-focused package; the issue notes that this may require significant refactoring or code generation.

Written by the indexing model from the issue text.