exposure – For the same FOV and f-stop, will total luminous flux increase linearly with sensor area?

Suppose I have two different cameras, one with a sensor whose width is twice the other’s. Further assume the sensors have the same aspect ratio.

Suppose I put 35 mm-equivalent lenses on both, thus giving the same FOV (field of view).

If they both have the same f-stop, would the luminous flux on the larger sensor be 2^2 = 4x that on the smaller sensor?

I.e., for the same FOV and f-stop, would the large-sensor camera only need half the exposure time to achieve the same EV as the small-sensor camera?*

And would that in turn mean that, for the same FOV and f-stop, the aperture diameter of the lens on the large-sensor camera would need to be twice that on the small-sensor camera?

Finally, how would their depth of fields compare?

*This answer said that, for the same f-stop and FOV, the images falling on the sensors would be equally bright (How does sensor size affect the brightness of an image given the same field of view and aperture?), but didn’t precisely define brightness. I think by brightness they meant lumens per unit area (even though brightness is actually a perceptual term), not total luminous flux. If so, then I believe what I wrote in italics would be correct.