Related to, yes. Governed by, no. The devil is in the details, and he's a jealous bastard.

For instance, nothing in the physical realm "gathers" light, except for a black hole, and I for one am not sticking a gravitational singularity in front of my eye... Yes, a great deal of the misunderstandings revolve around terminology; this is an area where words/semantics matter.

A larger objective allows more light to transmit, though some will be reflected, some refracted, and some absorbed by the material. The light that DOES transmit has the same thing happen with every single layer it passes through, so any light "lost" is usually to being absorbed by the medium through which it transmits. This is where well-made, well-cut, well-polished glass plays more of a role, to my mind, than the size of the tube, once the light is past the objective. The difference between a 1" and 30mm tube, in this regard, is evident only to instrumentation and afficionados, but arguably may be their center of gravity between choosing between the two.

EDIT: Also, don't forget, a giant part of the "brightness" factor is from the optic subjectively moving your eye closer to what you're observing. Pretty much anything with magnification is going to "brighten" a scene. I think of the other stuff as refining that most basic premise, but that's open to interpretation and off-topic.....