Looks like everyone glossed over your post, but I'm going to re-highlight this.
A long (loonngg...) time ago when I was studying for something (PPL), I read this paragraph in the AIM (emphasis are mine):
VFR Flights.
2. Departing aircraft require a clearance to depart Class B airspace and should advise the clearance delivery position of their intended altitude and route of flight. ATC will normally advise VFR aircraft when leaving the geographical limits of the Class B airspace. Radar service is not automatically terminated with this advisory unless specifically stated by the controller.
In my young misreading of this paragraph, I took it to mean that you needed to be cleared
into and
out of Class B airspace. Now that I reread it 30 years later, I don't think it means what I thought it did. That second part about advising CD makes it sounds like this is only applicable on the ground and departing an airfield in the Class B itself. I'm sure there were a number of New York controllers that rolled their eyes when I was coming up on the lateral confines of the Class B and called them to confirm that I was cleared out of the B. There no need for the exit clearance if you are just transiting. Am I reading that correctly? I'm basing this on the fact that the next paragraph says:
3. Aircraft not landing or departing the primary airport may obtain an ATC clearance to transit the Class B airspace when traffic conditions permit and provided the requirements of 14 CFR Section 91.131 are met. Such VFR aircraft are encouraged, to the extent possible, to operate at altitudes above or below the Class B airspace or transit through established VFR corridors. Pilots operating in VFR corridors are urged to use frequency 122.750 MHz for the exchange of aircraft position information.