Best way to debug email deliverability when background jobs stall?

Context: I'm working on mlops 040 and ran into a decision point.

• What I’ve tried: basic setup + quick benchmarks.
• Constraints: limited time, want something stable.

Question: Best way to debug email deliverability when background jobs stall?

Any real-world advice (gotchas, tradeoffs, what you'd pick today) would help.