Just out of curiosity, Boar, how did you determine it flys in the face of the OED, which says (your excerpt) that boudin is a probable, but not proven, origin: The identity of the words, though highly probable, cannot however be held to be proved,?
Earlier in the same excerpt it says they are likely the same word: Note. ME. poding, mod. pudding, and F. †bodin, boudin, have so many points in common that, but for the difficulties of form, they would at once be identified as the same word.
So, the OED seems to support the notion that boudin might be the root.
Moving away from the dictionary, in the late 18th and early 19th centuries (and perhaps both earlier and later), boudin refered, in common usage, to the intenstines of an animal and to sausages that used them as casings. Indeed, in Lousiana, there is still a popular sausage made of pork, pork liver, rice, and seasonings that is called boudin. It's not hard to see a connection between the sausage-puddings of an earlier day and anything made in a similar manner.
As a matter of historical interest, the second favorite food of the Mountain Men, during the American Fur Trade, was to open the body cavity of a freshly killed buffalo. Two Mountain Men would start on either side of the gut, which they called boudins, gulping them down as fast as possible.
One possible contribution to the popularity of these boudins is that the only vegetables (well, fiber anyway) those people ate was the green matter in the intestines.
Pudding is one of those strange words anyway, one which is used to describe disparate dishes. On one hand, savory sausages can be puddings. But so, too, are creamy sweets. And most authorities take a dish like luchen kugle and translate it as noodle pudding---which, in form, is a long way from either of the other two.