I'm not sure that the 7110.65 and AIM are really contradictory. The controller is required to coordinate the transition through a class B, C, or D surface area only if the controller issues a clearance that will require the aircraft to enter that airspace.
Ref 7110.65: 2-1-16. SURFACE AREAS
a. Coordinate with the appropriate nonapproach control tower on an individual aircraft basis before issuing a clearance which would require flight within a surface area for which the tower has responsibility unless otherwise specified in a letter of agreement.
FAAO JO 7210.3, Para 4-3-1, Letters of Agreement.
14 CFR Section 91.127, Operating on or in the Vicinity of an Airport in Class E Airspace.
P/CG Term- Surface Area.
b. Coordinate with the appropriate control tower for transit authorization when you are providing radar traffic advisory service to an aircraft that will enter another facility's airspace.
The pilot is not expected to obtain his/her own authorization through each area when in contact with a radar facility.
Well as I said "semantically" (if that's a word) the AIM and .65 do not contradict however in when you try to apply the two is where the problem comes. I'll come back to that.
In your own post you quote paragraph "b" which is the most applicable as I see it because it's the one that the note applies to.
"b. Coordinate with the appropriate control tower
for transit authorization
when you are providing radar traffic advisory service
to an aircraft that will enter another facility's airspace."
It doesn't say anything about issuing a clearance. Then comes the note to making more clear about what the controller is expected to do.
The pilot is not expected to obtain his/her own authorization through each area when in contact with a radar facility."
Back to the application of the AIM and .65's instructions. If the ATC handbook is telling controllers that pilots aren't expected to do the coordination, then it would make sense that the controllers are the one who are expected to (they're the other party involved in this exchange). And it's hard to imagine that a controller would be expected to do something without being held somehow responsible for doing it.
On the other side of the the radio is the pilot. If an aircraft is receiving flight following from a radar facility and it's flight path it taking it though the D the AIM states it's their responsibility to meet the requirements of entry into controlled airspace. However that would require them to possibly leave the radar facility's frequency, which you can't do without permission, or be actively monitoring two different ATC facilities at the same time which could be very taxing. Then what if the tower gives the aircraft a restriction or instruction which is on conflict an instruction to the radar facility gives or vice versa. That could lead to one big mess. All around it puts the pilot between a rock and hard place
I'll just add one more thing. I don't think this part of your statement is wrong:
the pilot is responsible for obtaining authorization to transition any surface area and is not implicitly cleared through that airspace just by virtue of receiving flight following.
The AIM is pretty clear on that and while I didn't look up the FAR it referenced I'm guessing it says the same. But I don't think there is any differentiation between in the controller's responsibility in when a clearance as been issued and simple flight following is being given.