When it comes to the overall quality of the image, counting both the size of the sensor and the absolute diameter of the aperture is counting twice. The amount of light that hits the sensor, for a given field of view, is entirely determined by the absolute diameter of the aperture. Bigger sensors collect more light because you can use a bigger absolute aperture for a given f-stop and field of view. Or if you prefer, a larger absolute aperture collects more light for a given f-stop and field of view because you can use a bigger sensor.

Obviously, the format has implications for DoF, but those can be either positive or negative. When I shoot 4x5, I spend most of my time fiddling around with movements trying to get everything in focus :)

Exactly. The thing that really matters for light gathering is Etendue -- the product of the area of the source and the solid angle that the system's entrance pupil subtends as seen from the source -- which is conserved (in a lossless optical system).

[1] https://en.wikipedia.org/wiki/Etendue

There's more nuance than that, because sensors have a certain fixed amount of area, given a resolution, dedicated to control circuitry. Larger sensors make the control circuitry a smaller fraction of overall area.

The focal length of the lens and the aperture diameter is what controls depth of field. The sensor size doesn't actually matter; you can crop a photo later (which is like using a smaller sensor) and the depth of field doesn't change. The way it gets involved is that a certain focal length with a certain sensor size will result in objects in the photograph being a certain size. That means that a 35mm camera will use a 50mm "normal" lens, and a 4x5 camera will use a 150mm "normal" lens. Attach a 50mm lens to a 35mm camera, take a picture, replace the camera with a 150mm lens on a 4x5 camera, and objects in each photograph will be about the same size. But, the 150mm lens will obviously produce more blurring away from the focal plane; that's just what longer lenses do.

The iPhone's telephoto lens has a 6mm focal length, and so tends to produce an image with more in focus than a lens that's 25x longer.

(Why do phones use a 6mm lens and not a 150mm lens? Because they don't have 150mm to spare between the lens and the sensor.)

To add to what you said, the depth of field calculation relies on the circle-of-confusion that you pick (which is based on the viewing distance and the size of final reproduction), and is therefore also closely dependent on the resolving power of the sensor, and thus pixel density. In addition DoF also depends on the focusing distance.

>(Why do phones use a 6mm lens and not a 150mm lens? Because they don't have 150mm to spare between the lens and the sensor.)

That is not true for today's lens formulas. You're probably thinking of old singlet designs from centuries back. For e.g. reverse-telephoto lens designs allows you to make lens much shorter than their focal length. Its all about getting the final magnification by positioning positive and negative lens elements, and ofcource other corrective lenses appropriately in the lens formula. This is why you can use "tele-converters" on lenses to increase their magnification (and thus their effective focal length)