Appease Clippy 1.68 and refactor http_api (#4068)

## Proposed Changes

Two tiny updates to satisfy Clippy 1.68

Plus refactoring of the `http_api` into less complex types so the compiler can chew and digest them more easily.

Co-authored-by: Michael Sproul <michael@sigmaprime.io>

testing/execution_engine_integration/src/main.rs

@@ -1,4 +1,3 @@
-#![recursion_limit = "1024"]
 /// This binary runs integration tests between Lighthouse and execution engines.
 ///
 /// It will first attempt to build any supported integration clients, then it will run tests.