BTACON
I believe you are quite right about verifying poisition prior to takeoff clearance, I read the manual too and you quote correctly.
There will be plenty of blame to go around. I do not mean this to be an idictment of the pilots or controller.
I do mean this as an indictment of the system. the system that has pilots not checking compass on runway as naturally as scratching your nose when it itches. the system that has 1 controller in the tower at all, let alone working a local position as well as radar (probably on a brite scope at that!).
and all runways that are closed, even just part time should have a Big X on them and illuminated at night.
funny thing, at LGA, there are magnetic annomalies which would show line up on the runway incorrectly...in fact slewing the HSI to runway heading on lineup is the norm if you want to fly right.