Yeet env_logger into the sun (#7872)

- Remove explicit `env_logger` usage from `state_processing` tests and `lcli`.
- Set up tracing correctly for `lcli` (I've checked that we can see logs after this change).
- I didn't do anything to set up logging for the `state_processing` tests, as these are rarely run manually (they never fail). We could add `test_logger` in there on an as-needed basis.

lcli/Cargo.toml

@@ -20,7 +20,6 @@ bls = { workspace = true }
 clap = { workspace = true }
 clap_utils = { workspace = true }
 deposit_contract = { workspace = true }
-env_logger = { workspace = true }
 environment = { workspace = true }
 eth2 = { workspace = true }
 eth2_network_config = { workspace = true }