Thank you for taking the time to bring these items to our attention. Over the years I have had occassion to ask folks to please demonstrate with genealogical proofs any Van Sickle[n] family they have located that "dropped the Van." Because I haven't run across any does not mean, naturally enough, that none exist. I just haven't found any. And, also quite naturally, I will be glad to hear now from anyone who might be able to prove a case, or cases, where this has happened.
Conversely, there are demonstrable cases of Sickles/Sickels descendants who acquired the "Van". Thus the statement in the obituary you have kindly provided that General Sickles' grandfather used the "Van" is not especially persuasive, to me at least, that Gen. Sickles was a descendant of Ferdinandus Van Sickelen.
One may get a better view of the potentialities, I think, with a visit to the Sickles GenForum, which includes more than one thread on the General.