Does lens front element size matter?

[ReinisK]

Apart from cost, weight and resolution goals and differences for each lens, what else could be the difference between lenses with bigger and smaller front element?

Everyone's talking about the 3d look of full frame cameras vs crop cameras, but I haven't really found an explanation from a physics POV. So I came up with a guessing, that the full frame sensor gives more of a 3d look, because being bigger, it kind of sees more around things, than a smaller sensor. Like it can see more of subjects sides, because it is wider, and the plane is bigger to the subject. The difference should be bigger and more visible using shorter FLs. Basically, it would be the same thing as a soft light source (bigger plane) vs a hard light source (small plane) where the bigger planes very side sees more around the subject.

So getting back to lenses' front elements - would it be possible, that the lenses with a bigger front element, also see more around the subject, therefore, could create a more 3d look? For someone, who is good at physics and knows optics it probably could be an easy question, I guess?

[no_connection]

Front element size together with focal length determine the maximum aperture of the lens. Divide the focal length by the aperture and you get front element diameter, give or take some tolerance. Front element diameter determine the depth of field together with field of view and how you view the resulting image. It's like magic how they all fit together.

[Nikkor]

What I have noticed is that lenses with large front elements have less vignetting, so the dof is not only shallow in the center but also in the corners. (Compare shots in a classic 50 1.4, the corners will be as dark and dof won't change when stopping down to f2.8.