Improve execution engine error logging

It seems that we don't emit any error log if EL is offline a lot of the time, eg. after start, we just have a log that states it's offline but not what exact error caused this.

Before Electra, we had the Eth1Provider that at least emitted a error log but now this is no longer the case due to EIP-6110 since we disabled Eth1 data polling.

Start by tracing the execution engine's offline handling and compare it with the former Eth1Provider error logging path. Read the EIP-6110-related changes that disabled Eth1 data polling, then verify that the underlying execution-layer error is logged when the EL is offline, including after startup.

Written by the indexing model from the issue text.