Part of the problem with it remaining culturally (and hopefully linguistically) Dutch is one of numbers. English colonists kept encroaching into the Dutch-claimed area before the English took control, and began settling in much larger numbers once it was legally English. Unless you can boost the number of Dutch settlers in the area and have the English authorities on either side (i.e. New England and 'greater' Virginia) co-operate with New Netherland's authorities in keeping anglophone settlers out, the area is likely to be swamped by English Americans.
Also, the colony was never seen as an end in itself by the WIC (Dutch West India Company) and the States-General. Its main purposes were to support fur trapping further up the Hudson valley, and to provide a victualling station for WIC privateers returning from the Caribbean to the Netherlands. I feel you'd need backing, both political and financial, from the Dutch nobility before New Netherland could boast de facto as well as de jure control over the area it claimed.
That said, however, if you could keep New Netherland as a going concern, it will most definitely lead to a division of the English colonies into two separate cultural units. Even OTL there are distinct cultures in New England, the ex-New Netherland states, and the Upper and Deep South (Virginia southwards). With a separate polity acting as a bulwark between New England and 'Chesapeake-Bay-southwards', I honestly can't see a single American identity developing. Far more likely you'd end up with at least two: New Englander and Virginian/Columbian/some other catch-all term for the southern colonies/states.
I reckon that it's unlikely a surviving New Netherland would prevent major rebellions in the anglophone colonies, but I think it's likely they'd play out very differently. Major factors would be who held New Netherland, whether the dutch-speakers sided with or against any anglophone rebellions, and whether the two divided blocks of English colonies end up feeling enough of a common bond with each other to rebel together, or whether they'd view rebellion in one block as an opportunity to profit at that block's expense.
The trigger for any rebellions would also be important. If you have a major Virginian/southern rebellion triggered by something slave-related, for example the legal judgment in 1773(?) that made slave-holding illegal in England, then I imagine New England would be far less likely to rise with the southerners as their economy depended far less on slave-holding, IIRC.
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PS: TL already under construction, it's just taking a while to filter out the handwavium
