There was a Buckshutem Branch in Monmouth County by Wykoff Mills, now called Bannen Meadow Branch (Brook), so that name is not unique to Cumberland County (Buckshootem Bridge in Beekman 1915: 116–117). It is in what is now Howell Township, named in honor of Governor Richard Howell of Shiloh, NJ, although I'm not sure he ever lived in Howell to bestow such a Cumberland County namesake.
My sense is that the channel crossings are crossways built for forest product extraction by the early eighteenth century, especially for Atlantic whitecedar—as seen on Deep Run on Hartman's Map 5:
As I understand crossways were corduroy roads that linked hummocks as they crisscrossed a stream's broad shallow paleochannel. Some hummocks are remnants of windblown sands, sands that choked Buckshutem's channel (Manumuskin's "no channel to speak of"). Big beaver dams are often associated with beaver ponds that are of a distinctive shape (above left), like a riverine spung. However, it would not be a far fetch to assume that some crossways without those ponds were initiated by beaver activity. Permafrost hydrologist Hok Woo writes about beaver activity into Polar regions, so these critters have likely been here in the Pine Barrens for over 14,000 years—into the end of the ice age. We don't know if their kin the Giant Beavers ever built dams, as they went extinct at Pleistocene's end.
Break-dams, too, were built to impound floating pools to flood winter-cut logs down to the saw mill during spring high water. Beaver knew the best places to build dams, and people followed.