Does an f2.8 (say) lens collect up more light than an f.5.6 lens through its design geometry or is correct exposure independent of lens?

This prompted by the fact that my Dunnock shot about this time last year
http://www.worldphotographyforum.com...4173&nocache=1
worked at f13 using sigma 50-150 2.8 lens giving adequate DOF at 150 about 1.2M away (its a crop of approx 50% width of the frame taken) in what I thought was terrible light, admittedly ISO800. Perhaps more light than I thought bounced around by snow??

Struggling now with ISO200 f9 1/250 400mm in very good light and not getting enough DOF for similar sized bird using Nikkor 80-400 at closest focus ie 2.3M

Related question if not already answered - what circumstances would f.2.8 or lower be useful for other than pankakes flat on?