[TRACING] Improved testing of tracing functionality

See https://github.com/awslabs/shuttle/pull/128, in particular: https://github.com/awslabs/shuttle/pull/128#discussion_r1419144606.

Resolving this will involve:
1. Creating a custom tracing `Subscriber` which checks the traces emitted by the tests in `tracing.rs` (and optionally elsewhere).
2. Un-`#[ignore]`-ing the tests in `tracing.rs`.
3. Make the tests in `tracing.rs` always enable some `LevelFilter` (as in: overwriting the level set by `RUST_LOG`). Bonus points for making the tests try all different `LevelFilter`s.
4. [OPTIONAL] Add more tests.

Start by reading the discussion on PR 128 and the tests in tracing.rs. Create the custom tracing Subscriber described there, remove the #[ignore] markers, and ensure the tests override RUST_LOG with a LevelFilter. Done means the tracing tests run reliably and check the traces they emit.

Written by the indexing model from the issue text.