Use efficient payload reconstruction for HTTP API (#4102)

## Proposed Changes

Builds on #4028 to use the new payload bodies methods in the HTTP API as well.

## Caveats

The payloads by range method only works for the finalized chain, so it can't be used in the execution engine integration tests because we try to reconstruct unfinalized payloads there.

testing/execution_engine_integration/src/test_rig.rs

@@ -626,9 +626,10 @@ async fn check_payload_reconstruction<E: GenericExecutionEngine>(
     ee: &ExecutionPair<E, MainnetEthSpec>,
     payload: &ExecutionPayload<MainnetEthSpec>,
 ) {
+    // check via legacy eth_getBlockByHash
     let reconstructed = ee
         .execution_layer
-        .get_payload_by_block_hash(payload.block_hash(), payload.fork_name())
+        .get_payload_by_hash_legacy(payload.block_hash(), payload.fork_name())
         .await
         .unwrap()
         .unwrap();