The sentence in question is:
At five stories high with a crew of 20, the cigar-shaped behemoth was grinding away underground on a two-mile-long, $3.1 billion highway tunnel under the city’s waterfront on Dec. 6 when it encountered something in its path that managers still simply refer to as “the object.
Simplified, the phrase in question is:
behemoth was grinding away on a tunnel on Dec. 6 when it
The first question to ask is: is 'grinding away' literally grinding (as the machine literally does) or is it the idiom or phrase 'grinding away' being used. The latter is indeed usually followed by 'at'. Perhaps it is both and intended as a mild pun.
In idiomatic usage, 'grinding away on a list of tasks' and 'grinding away in a job' is not unheard of. It appear that 'at' is used when it is is a single concrete object or task and 'on' and 'in' for more amorphous objects.
In the cited article, the thought may have been rephrased as 'grinding away at rock under the sea' and reading twice, 'grinding away on a tunnel' does not quit make sense, and is thus poor usage. However, on first reading, the sense is that of a tunnel project.
I think the apparent dissonance is due to the 'grinding' being quite literal in this case.
Lastly, the 'on' in 'on Dec. 6' is correct when used for a date.