Ensure safe usage of `RwLock`s are safe in `beacon_node_fallback`

[jimmygchen]

Description

#4393 introduces some new usage of RwLock in beacon_node_fallback. This ensures safe, concurrent access to node health data across threads. We've done a round of review on this and don't think there's any deadlock risk, or any locks being held for too long, however there are areas we could potentially simplify and improve on:

• Gather some test metrics to make sure requests are sent timely to BNs
• Scan the code with lockbud (Get lockbud running again #6413)
• Add extensive tests for beacon node fallback (WIP branch with some test rig)
• This review comment to simplify some logic

[eserilev]

fwiw lockbud isn't picking up any deadlocks in beacon_node_fallback. You can see the results in the lockbud CI step in this PR #6470

[jimmygchen]

lockbud has been integrated into our CI pipeline and has not found any lock issues in beacon_node_fallback. The new fallback mechanism was included in 6.0.0 and we have not noticed any regression. @chong-he has also added some new tests for beacon_node_fallback so I believe we can close this issue now.