The book Torching the Fink Books and Other Essays on Vernacular Culture (by Archie Green) offers seven pages on the origin of dutchman. In summary, the most plausible theory suggests that dutchman was first applied by American mechanics to compliment the skill of their German fellows around 1830s-1840s.
Here are some relevant parts from the book:
The first lexicographer to recognize dutchman as applied to a work process was John Russell Bartlett in the Dictionary of Americanisms: "A flow in a stone or marble slab, filled up by an insertion."
Bartlett recorded dutchman in his Dictionary in 1859. It is a good rule of thumb in language study that a word circulates orally many years before it gets into print, and often for some time in print before it is rewarded by being placed in a dictionary.
The first woodworkers in the New World were colonists and pioneers who felled trees, cut logs into lumber, and built their own crude implements. While the nation was still young, the invention of specialized woodworking machinery set the stage for factory production of wood products. It was only after the application of the steam engine to saw mills, and the introduction of the circular saw and related tools, that cheap, rapid, mass production of finished lumber, furniture, sashes and doors, and stock trim was possible. Hence it was in the decades of the 1830s and 1840s that mill-cabinet and building entrepreneurs reached out to Germany and other European areas for skilled help.
The early machine woodworking trade was centered in the growing cities of the East Coast, Boston, New York, and Philadelphia; then it gradually spread to Cincinnati, Chicago, and Grand Rapids as the forests of Wisconsin and Michigan were opened to lumbering operations.
Some place, some time, perhaps in an East Coast cabinet shop in the Age of Jackson, a native tradesman complimented the craftsmanship of a German mechanic and a new word was born. Neither lexicographer Bartlett nor shipwright Carrwardine is here to give his sources.
Source: Google Books