Puritan Anglican dispute seems possible - and would explain the nastiness - the feel isn't the same as, say, the tales of Merry Maidens turned to stone for dancing on the Sabbath.
The time of year and the readiness of the soil for ploughing was once of the oddities I noticed - if the carol did originate in the 17th century, that was during the Little Ice Age, and the ground would have been hard.
Another was the observer - "we" went to plough, but thereafter everything in third person, including Jesus speaking of himself.

Penny