Further remove EE redundancy (#3324)

## Issue Addressed

Resolves #3176

## Proposed Changes

Continues from PRs by @divagant-martian to gradually remove EL redundancy (see #3284, #3257).

This PR achieves:

- Removes the `broadcast` and `first_success` methods. The functional impact is that every request to the EE will always be tried immediately, regardless of the cached `EngineState` (this resolves #3176). Previously we would check the engine state before issuing requests, this doesn't make sense in a single-EE world; there's only one EE so we might as well try it for every request.
- Runs the upcheck/watchdog routine once per slot rather than thrice. When we had multiple EEs frequent polling was useful to try and detect when the primary EE had come back online and we could switch to it. That's not as relevant now.
- Always creates logs in the `Engines::upcheck` function. Previously we would mute some logs since they could get really noisy when one EE was down but others were functioning fine. Now we only have one EE and are upcheck-ing it less, it makes sense to always produce logs.

This PR purposefully does not achieve:

- Updating all occurances of "engines" to "engine". I'm trying to keep the diff small and manageable. We can come back for this.

## Additional Info

NA

beacon_node/beacon_chain/tests/payload_invalidation.rs

@@ -1003,6 +1003,11 @@ async fn payload_preparation_before_transition_block() {
     let rig = InvalidPayloadRig::new();
     let el = rig.execution_layer();
 
+    // Run the watchdog routine so that the status of the execution engine is set. This ensures
+    // that we don't end up with `eth_syncing` requests later in this function that will impede
+    // testing.
+    el.watchdog_task().await;
+
     let head = rig.harness.chain.head_snapshot();
     assert_eq!(
         head.beacon_block