I also remember
Yes; also. Because the cup provided additional memories, not competing ones.
getting mullered in the very next round, falling rapidly into the relegation zone. Any win over Man City was far overshadowed by the Great Escape that season.
Not for me. I barely remember the relegation fight (or the mullering). But that's beside the point. We didn't drop into danger because of the cup competition. We did that because we had a poor team and a poor manager.