I don't think the "stick in the mud" line is connected.
Where it's used in the references I'm looking at it means actually stuck in mud.
Here are the examples:
1. "...the beautiful sloughs where Billy Barlow will sometimes get stuck in the mud."
2. and, like "Billy Barlow" (of dear childhood's memory), stuck in the mud.

The first from a Civil War Diary. The second from a New Zealand newspaper.